Search: keyword:new
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A352816
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G.f. A(x) satisfies: 1 - x = Sum_{n>=0} x^n * (x^n + (-1)^n*A(x))^n.
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1, 2, 3, 7, 16, 40, 105, 281, 770, 2147, 6080, 17431, 50459, 147265, 432821, 1279978, 3806154, 11373668, 34136769, 102863360, 311065038, 943740309, 2871717550, 8762164066, 26801981013, 82172139800, 252470206882, 777243596670, 2397210487402, 7406340364950
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OFFSET
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0,2
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LINKS
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Table of n, a(n) for n=0..29.
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FORMULA
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G.f. A(x) satisfies:
(1) 1 - x = Sum_{n>=0} x^n * ( x^n + (-1)^n*A(x) )^n.
(2) 1 - x = Sum_{n>=0} x^(n*(n+1)) / ( 1 + (-1)^n*x^(n+1)*A(x) )^(n+1).
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 3*x^2 + 7*x^3 + 16*x^4 + 41*x^5 + 107*x^6 + 286*x^7 + 782*x^8 + 2179*x^9 + 6172*x^10 + 17702*x^11 + 51272*x^12 + ...
where
1 - x = 1 + x*(x - A(x)) + x^2*(x^2 + A(x))^2 + x^3*(x^3 - A(x))^3 + x^4*(x^4 + A(x))^4 + x^5*(x^5 - A(x))^5 + x^6*(x^6 + A(x))^6 + ...
Also,
1 - x = 1/(1 + x*A(x)) + x^2/(1 - x^2*A(x))^2 + x^6/(1 + x^3*A(x))^3 + x^12/(1 - x^4*A(x))^4 + x^20/(1 + x^5*A(x))^5 + ...
Specific values.
A(x) = 2 at x = 0.2515345800906597549860933755999098710430442043412...
A(x) = 3 at x = 0.3019496486281905495940467534583642873678976347045...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, x^m * (x^m + (-1)^m*Ser(A))^m ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, sqrtint(#A), x^((m+1)*m)/(1 + (-1)^m*x^(m+1)*Ser(A))^(m+1) ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
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CROSSREFS
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Cf. A352817, A317997.
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KEYWORD
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nonn,new
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AUTHOR
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Paul D. Hanna, Apr 05 2022
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STATUS
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approved
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A352878
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G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * (A(x) - x^n)^n.
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+0
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1, 1, 1, 4, 9, 28, 86, 265, 855, 2783, 9214, 30867, 104481, 356859, 1228064, 4254760, 14827717, 51944178, 182819043, 646128479, 2292209747, 8159722774, 29137365046, 104343145880, 374640974686, 1348388811894, 4863897209014, 17581361532197, 63673157211566
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OFFSET
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0,4
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COMMENTS
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Compare the g.f. to: C(x) = Sum_{n>=0} x^n * C(x)^n, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A0000108).
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LINKS
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Table of n, a(n) for n=0..28.
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FORMULA
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G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n * (A(x) - x^n)^n.
(2) A(x) = Sum_{n>=0} (-1)^n * x^(n*(n+1)) / (1 - x^(n+1)*A(x))^(n+1).
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EXAMPLE
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G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 9*x^4 + 28*x^5 + 86*x^6 + 265*x^7 + 855*x^8 + 2783*x^9 + 9214*x^10 + 30867*x^11 + 104481*x^12 + ...
where
A(x) = 1 + x*(A(x) - x) + x^2*(A(x) - x^2)^2 + x^3*(A(x) - x^3)^3 + x^4*(A(x) - x^4)^4 + x^5*(A(x) - x^5)^5 + x^6*(A(x) - x^6)^6 + ...
also,
A(x) = 1/(1 - x*A(x)) - x^2/(1 - x^2*A(x))^2 + x^6/(1 - x^3*A(x))^3 - x^12/(1 - x^4*A(x))^4 + x^20/(1 - x^5*A(x))^5 + ...
Specific values.
A(1/4) = 1.56854185594017483280286112046695218005165445150...
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PROG
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(PARI) {a(n) = my(A=1); for(i=1, n,
A = sum(m=0, n, x^m * (A - x^m)^m +x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, sqrtint(#A)+1, (-1)^m * x^(m*(m+1))/(1 - x^(m+1)*Ser(A))^(m+1) ), #A-1)); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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Cf. A352816, A317997.
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KEYWORD
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nonn,new
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AUTHOR
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Paul D. Hanna, Apr 06 2022
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STATUS
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approved
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A352856
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G.f. A(x) satisfies: 1 = Sum_{n>=0} (-x)^n * A(x)^n * A(x*A(x)^(2*n)).
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+0
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1, 1, 1, 3, 10, 42, 216, 1208, 7476, 50476, 365155, 2809512, 22877097, 196157406, 1762794051, 16541259592, 161597116528, 1639375229394, 17228899619932, 187162632393721, 2097600065319188, 24211313789364265, 287351810807343160, 3501527503646399390
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OFFSET
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0,4
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LINKS
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Table of n, a(n) for n=0..23.
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 1 = Sum_{n>=0} (-x)^n * A(x)^n * A(x*A(x)^(2*n)),
(2) 1 = Sum_{n>=0} a(n) * x^n / (1 + x*A(x)^(2*n+1)).
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EXAMPLE
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G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 42*x^5 + 216*x^6 + 1208*x^7 + 7476*x^8 + 50476*x^9 + 365155*x^10 + 2809512*x^11 + ...
where
(1) 1 = A(x) - x*A(x)*A(x*A(x)^2) + x^2*A(x)^2*A(x*A(x)^4) - x^3*A(x)^3*A(x*A(x)^6) + x^4*A(x)^4*A(x*A(x)^8) - x^5*A(x)^5*A(x*A(x)^10) + x^6*A(x)^6*A(x*A(x)^12) + ...
(2) 1 = 1/(1 + x*A(x)) + 1*x/(1 + x*A(x)^3) + 1*x^2/(1 + x*A(x)^5) + 3*x^3/(1 + x*A(x)^7) + 10*x^4/(1 + x*A(x)^9) + 42*x^5/(1 + x*A(x)^11) + 216*x^6/(1 + x*A(x)^13) + ... + a(n)*x^n/(1 + x*A(x)^(2*n+1)) + ...
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PROG
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(PARI) /* 1 = Sum_{n>=0} (-x)^n * A(x)^n * A(x*A(x)^(2*n)) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(n=0, #A-1, (-x)^n*Ser(A)^n*subst(Ser(A), x, x*Ser(A)^(2*n)) ), #A-1)); A[n+1]}
for(n=0, 31, print1(a(n), ", "))
(PARI) /* 1 = Sum_{n>=0} a(n) * x^n / (1 + x*A(x)^(2*n+1)) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(n=0, #A-1, A[n+1]*x^n/(1 + x*Ser(A)^(2*n+1)) ), #A-1)); A[n+1]}
for(n=0, 31, print1(a(n), ", "))
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CROSSREFS
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Cf. A352853, A352854, A352855.
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KEYWORD
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nonn,new
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AUTHOR
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Paul D. Hanna, Apr 05 2022
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STATUS
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approved
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A352855
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G.f. A(x) satisfies: 1 = Sum_{n>=0} (-x)^n * A(x)^(3*n) * A(x*A(x)^n).
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1, 1, 3, 13, 69, 418, 2785, 19965, 151839, 1213329, 10117007, 87572771, 783819258, 7231777871, 68607958358, 667912243459, 6661081903672, 67956552527140, 708354657022645, 7536107197624606, 81756574537050031, 903706858640897865, 10170700352770130132
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OFFSET
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0,3
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LINKS
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Table of n, a(n) for n=0..22.
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 1 = Sum_{n>=0} (-x)^n * A(x)^(3*n) * A(x*A(x)^n),
(2) 1 = Sum_{n>=0} a(n) * x^n / (1 + x*A(x)^(n+3)).
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 69*x^4 + 418*x^5 + 2785*x^6 + 19965*x^7 + 151839*x^8 + 1213329*x^9 + 10117007*x^10 + ...
where
(1) 1 = A(x) - x*A(x)^3*A(x*A(x)) + x^2*A(x)^6*A(x*A(x)^2) - x^3*A(x)^9*A(x*A(x)^3) + x^4*A(x)^12*A(x*A(x)^4) - x^5*A(x)^15*A(x*A(x)^5) + x^6*A(x)^18*A(x*A(x)^6) + ...
(2) 1 = 1/(1 + x*A(x)^3) + 1*x/(1 + x*A(x)^4) + 3*x^2/(1 + x*A(x)^5) + 13*x^3/(1 + x*A(x)^6) + 69*x^4/(1 + x*A(x)^7) + 418*x^5/(1 + x*A(x)^8) + 2785*x^6/(1 + x*A(x)^9) + ... + a(n)*x^n/(1 + x*A(x)^(n+3)) + ...
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PROG
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(PARI) /* 1 = Sum_{n>=0} (-x)^n * A(x)^(3*n) * A(x*A(x)^n) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(n=0, #A-1, (-x)^n*Ser(A)^(3*n)*subst(Ser(A), x, x*Ser(A)^n) ), #A-1)); A[n+1]}
for(n=0, 31, print1(a(n), ", "))
(PARI) /* 1 = Sum_{n>=0} a(n) * x^n / (1 + x*A(x)^(n+3)) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(n=0, #A-1, A[n+1]*x^n/(1 + x*Ser(A)^(n+3)) ), #A-1)); A[n+1]}
for(n=0, 31, print1(a(n), ", "))
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CROSSREFS
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Cf. A352853, A352854, A352856.
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KEYWORD
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nonn,new
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AUTHOR
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Paul D. Hanna, Apr 05 2022
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STATUS
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approved
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A352850
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E.g.f. A(x) satisfies: [x^k] A(x)^n = 0 for k = n*(n-1)/2 + 1 through n*(n+1)/2, for n > 1.
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1, 1, -1, 3, -18, 150, -1590, 20790, -321930, 5751270, -116263350, 2628687600, -65723427000, 1800165767400, -53590470733800, 1722471401851200, -59462981199121500, 2194469988757546500, -86216322855144982500, 3592621186732990912500, -158253385898375658891000
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OFFSET
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0,4
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LINKS
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Table of n, a(n) for n=0..20.
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EXAMPLE
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E.g.f.: A(x) = 1 + x - x^2/2! + 3*x^3/3! - 18*x^4/4! + 150*x^5/5! - 1590*x^6/6! + 20790*x^7/7! - 321930*x^8/8! + 5751270*x^9/9! - 116263350*x^10/10! + ...
The series of reduced fractional coefficients begins:
A(x) = 1 + x - 1/2*x^2 + 1/2*x^3 - 3/4*x^4 + 5/4*x^5 - 53/24*x^6 + 33/8*x^7 - 511/64*x^8 + 3043/192*x^9 - 4101/128*x^10 + ...
Related table.
The table of coefficients of x^k/k! in A(x)^n begins:
n=1: [1, 1, -1, 3, -18, 150, -1590, 20790, -321930, 5751270, ...];
n=2: [1, 2, 0, 0, -6, 60, -660, 9240, -149100, 2729160, ...];
n=3: [1, 3, 3, -3, 0, 0, 0, 1260, -28350, 595350, -13012650, ...];
n=4: [1, 4, 8, 0, -12, 0, 120, 0, 0, 0, 0, 30353400, -1166734800, ...];
n=5: [1, 5, 15, 15, -30, -30, 150, 1050, -1050, -47250, 1445850, 0, 0, 0, 0, 0, -141443655853500, ...];
...
which illustrates that A(x)^n has n contiguous zeros for coefficients of x^k starting at k = n*(n-1)/2 + 1 through k = n*(n+1)/2, for n > 1.
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PROG
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(PARI) {a(n) = my(A=1+x, m=floor(sqrt(2*n)+1/2));
for(k=1, m, A = ( truncate(A^k) + x*O(x^(k*(k+1)/2)) )^(1/k));
n!*polcoeff(H=A, n)}
for(n=0, 21, print1(a(n), ", "))
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KEYWORD
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sign,new
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AUTHOR
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Paul D. Hanna, Apr 05 2022
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STATUS
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approved
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A352854
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G.f. A(x) satisfies: 1 = Sum_{n>=0} (-x)^n * A(x)^(2*n) * A(x*A(x)^n).
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+0
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1, 1, 2, 6, 23, 103, 516, 2819, 16517, 102615, 670503, 4580064, 32553887, 239884108, 1827188093, 14351353937, 115997378072, 963164672275, 8203632154685, 71582150287243, 639150768866594, 5833969369906384, 54387003936658041, 517419092989229133
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OFFSET
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0,3
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LINKS
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Table of n, a(n) for n=0..23.
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 1 = Sum_{n>=0} (-x)^n * A(x)^(2*n) * A(x*A(x)^n),
(2) 1 = Sum_{n>=0} a(n) * x^n / (1 + x*A(x)^(n+2)).
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 23*x^4 + 103*x^5 + 516*x^6 + 2819*x^7 + 16517*x^8 + 102615*x^9 + 670503*x^10 + 4580064*x^11 + ...
where
(1) 1 = A(x) - x*A(x)^2*A(x*A(x)) + x^2*A(x)^4*A(x*A(x)^2) - x^3*A(x)^6*A(x*A(x)^3) + x^4*A(x)^8*A(x*A(x)^4) - x^5*A(x)^10*A(x*A(x)^5) + x^6*A(x)^12*A(x*A(x)^6) + ...
(2) 1 = 1/(1 + x*A(x)^2) + 1*x/(1 + x*A(x)^3) + 2*x^2/(1 + x*A(x)^4) + 6*x^3/(1 + x*A(x)^5) + 23*x^4/(1 + x*A(x)^6) + 103*x^5/(1 + x*A(x)^7) + 516*x^6/(1 + x*A(x)^8) + ... + a(n)*x^n/(1 + x*A(x)^(n+2)) + ...
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PROG
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(PARI) /* 1 = Sum_{n>=0} (-x)^n * A(x)^(2*n) * A(x*A(x)^n) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(n=0, #A-1, (-x)^n*Ser(A)^(2*n)*subst(Ser(A), x, x*Ser(A)^n) ), #A-1)); A[n+1]}
for(n=0, 31, print1(a(n), ", "))
(PARI) /* 1 = Sum_{n>=0} a(n) * x^n / (1 + x*A(x)^(n+2)) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(n=0, #A-1, A[n+1]*x^n/(1 + x*Ser(A)^(n+2)) ), #A-1)); A[n+1]}
for(n=0, 31, print1(a(n), ", "))
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CROSSREFS
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Cf. A352853, A352855, A352856.
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KEYWORD
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nonn,new
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AUTHOR
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Paul D. Hanna, Apr 05 2022
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STATUS
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approved
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A352853
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G.f. A(x) satisfies: 1 = Sum_{n>=0} (-x)^n * A(x)^n * A(x*A(x)^n).
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+0
0
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1, 1, 1, 2, 5, 15, 53, 205, 865, 3928, 18943, 96387, 514745, 2871568, 16670197, 100400979, 625756254, 4026925835, 26705001158, 182188059474, 1276736262332, 9178023547010, 67597658864028, 509525556949153, 3926577535219879, 30908466065826275, 248308190295151020
(list;
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listen;
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OFFSET
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0,4
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LINKS
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Table of n, a(n) for n=0..26.
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 1 = Sum_{n>=0} (-x)^n * A(x)^n * A(x*A(x)^n),
(2) 1 = Sum_{n>=0} a(n) * x^n / (1 + x*A(x)^(n+1)).
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EXAMPLE
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G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 53*x^6 + 205*x^7 + 865*x^8 + 3928*x^9 + 18943*x^10 + 96387*x^11 + 514745*x^12 + ...
where
(1) 1 = A(x) - x*A(x)*A(x*A(x)) + x^2*A(x)^2*A(x*A(x)^2) - x^3*A(x)^3*A(x*A(x)^3) + x^4*A(x)^4*A(x*A(x)^4) - x^5*A(x)^5*A(x*A(x)^5) + x^6*A(x)^6*A(x*A(x)^6) + ...
(2) 1 = 1/(1 + x*A(x)) + 1*x/(1 + x*A(x)^2) + 1*x^2/(1 + x*A(x)^3) + 2*x^3/(1 + x*A(x)^4) + 5*x^4/(1 + x*A(x)^5) + 15*x^5/(1 + x*A(x)^6) + 53*x^6/(1 + x*A(x)^7) + ... + a(n)*x^n/(1 + x*A(x)^(n+1)) + ...
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PROG
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(PARI) /* 1 = Sum_{n>=0} (-x)^n * A(x)^n * A(x*A(x)^n) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(n=0, #A-1, (-x)^n*Ser(A)^n*subst(Ser(A), x, x*Ser(A)^n) ), #A-1)); A[n+1]}
for(n=0, 31, print1(a(n), ", "))
(PARI) /* 1 = Sum_{n>=0} a(n) * x^n / (1 + x*A(x)^(n+1)) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(n=0, #A-1, A[n+1]*x^n/(1 + x*Ser(A)^(n+1)) ), #A-1)); A[n+1]}
for(n=0, 31, print1(a(n), ", "))
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CROSSREFS
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Cf. A352854, A352855, A352856.
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KEYWORD
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nonn,new
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AUTHOR
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Paul D. Hanna, Apr 05 2022
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STATUS
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approved
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A352821
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G.f. A(x) satisfies: 1 - x = Sum_{n>=0} (x^(5*n) + (-1)^n*A(x))^n.
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+0
0
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1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 13, 21, 32, 46, 66, 99, 155, 244, 376, 569, 862, 1328, 2077, 3256, 5071, 7853, 12181, 19023, 29882, 47004, 73808, 115757, 181785, 286323, 452111, 714548, 1129185, 1784586, 2823069, 4472449, 7094472, 11261549, 17882350
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OFFSET
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1,5
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LINKS
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Table of n, a(n) for n=1..43.
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FORMULA
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G.f. A(x) satisfies:
(1) 1 - x = Sum_{n>=0} ( x^(5*n) + (-1)^n*A(x) )^n.
(2) 1 - x = Sum_{n>=0} x^(5*n^2) / (1 + (-1)^n*x^(5*n)*A(x))^(n+1).
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EXAMPLE
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G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 6*x^9 + 8*x^10 + 13*x^11 + 21*x^12 + 32*x^13 + 46*x^14 + 66*x^15 + ...
where
1 - x = 1 + (x^5 - A(x)) + (x^10 + A(x))^2 + (x^15 - A(x))^3 + (x^20 + A(x))^4 + (x^25 - A(x))^5 + (x^30 + A(x))^6 + ...
Also,
1 - x = 1/(1 + A(x)) + x^5/(1 - x^5*A(x))^2 + x^20/(1 + x^10*A(x))^3 + x^45/(1 - x^15*A(x))^4 + x^80/(1 + x^20*A(x))^5 + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, (x^(5*m) + (-1)^m*x*Ser(A))^m ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, sqrtint(#A\5), x^(5*m^2)/(1 + (-x)^(5*m)*x*Ser(A))^(m+1) ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
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CROSSREFS
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Cf. A317997, A352818, A352819, A352820.
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KEYWORD
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nonn,new
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AUTHOR
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Paul D. Hanna, Apr 05 2022
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STATUS
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approved
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A352820
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G.f. A(x) satisfies: 1 - x = Sum_{n>=0} (x^(4*n) + (-1)^n*A(x))^n.
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+0
0
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1, 1, 1, 2, 3, 4, 5, 7, 12, 20, 31, 48, 78, 131, 217, 354, 581, 971, 1634, 2739, 4580, 7699, 13027, 22092, 37449, 63551, 108176, 184637, 315530, 539625, 924125, 1585371, 2723675, 4683890, 8062277, 13892645, 23966392, 41384842, 71522034, 123706840, 214148865
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,4
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LINKS
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Table of n, a(n) for n=1..41.
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FORMULA
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G.f. A(x) satisfies:
(1) 1 - x = Sum_{n>=0} ( x^(4*n) + (-1)^n*A(x) )^n.
(2) 1 - x = Sum_{n>=0} x^(4*n^2) / (1 + (-1)^n*x^(4*n)*A(x))^(n+1).
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EXAMPLE
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G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 7*x^8 + 12*x^9 + 20*x^10 + 31*x^11 + 48*x^12 + 78*x^13 + 131*x^14 + ...
where
1 - x = 1 + (x^4 - A(x)) + (x^8 + A(x))^2 + (x^12 - A(x))^3 + (x^16 + A(x))^4 + (x^20 - A(x))^5 + (x^24 + A(x))^6 + ...
Also,
1 - x = 1/(1 + A(x)) + x^4/(1 - x^4*A(x))^2 + x^16/(1 + x^8*A(x))^3 + x^36/(1 - x^12*A(x))^4 + x^64/(1 + x^16*A(x))^5 + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, (x^(4*m) + (-1)^m*x*Ser(A))^m ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, sqrtint(#A\4), x^(4*m^2)/(1 + (-x^4)^m*x*Ser(A))^(m+1) ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
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CROSSREFS
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Cf. A317997, A352818, A352819, A352821.
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KEYWORD
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nonn,new
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AUTHOR
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Paul D. Hanna, Apr 05 2022
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STATUS
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approved
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A352819
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G.f. A(x) satisfies: 1 - x = Sum_{n>=0} (x^(3*n) + (-1)^n*A(x))^n.
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+0
0
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1, 1, 2, 3, 4, 6, 11, 19, 33, 60, 110, 201, 372, 696, 1305, 2456, 4654, 8855, 16896, 32366, 62217, 119910, 231704, 448879, 871531, 1695541, 3305057, 6454033, 12624041, 24731009, 48520396, 95324187, 187517959, 369329721, 728262059, 1437591201, 2840751293
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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LINKS
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Table of n, a(n) for n=1..37.
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FORMULA
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G.f. A(x) satisfies:
(1) 1 - x = Sum_{n>=0} ( x^(3*n) + (-1)^n*A(x) )^n.
(2) 1 - x = Sum_{n>=0} x^(3*n^2) / (1 + (-1)^n*x^(3*n)*A(x))^(n+1).
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EXAMPLE
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G.f.: A(x) = x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 11*x^7 + 19*x^8 + 33*x^9 + 60*x^10 + 110*x^11 + 201*x^12 + 372*x^13 + 696*x^14 + ...
where
1 - x = 1 + (x^3 - A(x)) + (x^6 + A(x))^2 + (x^9 - A(x))^3 + (x^12 + A(x))^4 + (x^15 - A(x))^5 + (x^18 + A(x))^6 + ...
Also,
1 - x = 1/(1 + A(x)) + x^3/(1 - x^3*A(x))^2 + x^12/(1 + x^6*A(x))^3 + x^27/(1 - x^9*A(x))^4 + x^48/(1 + x^12*A(x))^5 + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, (x^(3*m) + (-1)^m*x*Ser(A))^m ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, sqrtint(#A\3), x^(3*m^2)/(1 + (-x)^(3*m)*x*Ser(A))^(m+1) ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
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CROSSREFS
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Cf. A317997, A352818, A352820, A352821.
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KEYWORD
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nonn,new
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AUTHOR
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Paul D. Hanna, Apr 05 2022
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STATUS
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approved
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