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A152537
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Convolution sequence: convolved with A000041 = powers of 2, (A000079).
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5
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1, 1, 1, 2, 4, 9, 18, 37, 74, 148, 296, 592, 1183, 2366, 4732, 9463, 18926, 37852, 75704, 151408, 302816, 605632, 1211265, 2422530, 4845060, 9690120, 19380241, 38760482, 77520964, 155041928, 310083856, 620167712, 1240335424, 2480670848, 4961341696, 9922683391
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history;
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OFFSET
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0,4
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COMMENTS
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Terms are very similar to those of A178841. - Georg Fischer, Mar 23 2019
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LINKS
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Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
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FORMULA
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Construct an array of rows such that n-th row = partial sums of (n-1)-th row
of A010815: (1, -1, -1, 0, 0, 1, 0, 1,...).
A152537 = sums of antidiagonal terms of the array.
The sequence may be obtained directly from the following set of operations:
Our given sequence = A000041: (1, 1, 2, 3, 5, 7, 11,...). Delete the first
"1" then consider (1, 2, 3, 5, 7, 11,...) as an operator Q which we write in reverse with 1,2,3,...terms for each operation. Letting R = the target sequence (1,2,4,8,...); we begin a(0) = 1, a(1) = 1, then perform successive
operations of: "next term in (1,2,4,...) - dot product of Q*R" where Q is
written right to left and R (the ongoing result) written left to right).
Examples: Given 4 terms Q, R, we have: (5,3,2,1) dot (1,1,1,2) = (5+3+2+2) =
12, which we subtract from 16, = 4.
Given 5 terms of Q,R and A152537, we have (7,5,3,2,1) dot (1,1,1,2,4) = 23
which is subtracted from 32 giving 9. Continue with analogous operations to generate the series.
a(n)=sum_{j=0..n} A010815(j)*2^(n-j). G.f.: A000079(x)/A000041(x) = A010815(x)/(1-2x), where A......(x) denotes the g.f. of the associated sequence. - R. J. Mathar, Dec 09 2008
a(n) ~ c * 2^n, where c = A048651 = 0.28878809508660242127889972192923078... - Vaclav Kotesovec, Jun 02 2018
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EXAMPLE
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a(5) = 9 = 32 - 23 = (32 - ((7,5,3,2,1) dot (1,1,1,2,4)))
(1,1,2,3) convolved with (1,1,1,2) = 8, where (1,1,2,3...) = the first four partition numbers.
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MATHEMATICA
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nmax = 40; CoefficientList[Series[Product[1-x^k, {k, 1, nmax}] / (1-2*x), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 02 2018 *)
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PROG
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(PARI) /* computation by definition (division of power series) */
N=55;
A000079=vector(N, n, 2^(n-1));
S000079=Ser(A000079);
A000041=vector(N, n, numbpart(n-1));
S000041=Ser(A000041);
S152537=S000079/S000041;
A152537=Vec(S152537) /* show terms */ /* Joerg Arndt, Feb 06 2011 */
(PARI) /* computation using power series eta(x) and 1/(1-2*x) */
x='x+O('x^55); S152537=eta(x)/(1-2*x);
A152537=Vec(S152537) /* show terms */ /* Joerg Arndt, Feb 06 2011 */
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CROSSREFS
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Cf. A010815, A000041, A000079, A152538, A178841.
Sequence in context: A282956 A282986 A056185 * A182028 A081253 A118255
Adjacent sequences: A152534 A152535 A152536 * A152538 A152539 A152540
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KEYWORD
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nonn
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AUTHOR
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Gary W. Adamson, Dec 06 2008
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STATUS
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approved
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