0,5

The boustrophedon transform {t} of a sequence {s} is given by t_n = Sum_{k=0..n} T(n,k)*s(k). Triangle may be called the boustrophedon triangle.

The 'signed version' of the triangle is the exponential Riordan array [sech(x)+tanh(x), x]. - Peter Luschny, Jan 24 2009

Up to signs, the matrix is self-inverse: T^(-1)(n,k) = (-1)^(n+k)*T(n,k). - R. J. Mathar, Mar 15 2013

Reinhard Zumkeller, Rows n = 0..125 of table, flattened

Peter Luschny, The Swiss-Knife polynomials.

J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).

Wikipedia, Boustrophedon transform

Index entries for sequences related to boustrophedon transform

Sum_{k>=0} T(n, k) = A000667(n).

Sum_{k>=0} T(2n, 2k) = A000795(n).

Sum_{k>=0} T(2n, 2k+1) = A009747(n).

Sum_{k>=0} T(2n+1, 2k) = A003719(n).

Sum_{k>=0} T(2n+1, 2k+1) = A002084(n).

Sum_{k>=0} T(n, 2k) = A062272(n).

Sum_{k>=0} T(n, 2k+1) = A062161(n).

Sum_{k>=0} (-1)^(k)*T(n, k) = A062162(n). - Johannes W. Meijer, Apr 20 2011

E.g.f.: exp(x*y)*(sec(x)+tan(x)). - Vladeta Jovovic, May 20 2007

T(n,k) = 2^(n-k)C(n,k)|E(n-k,1/2)+E(n-k,1)|-[n=k] where C(n,k) is the binomial coefficient, E(m,x) are the Euler polynomials and [] the Iverson bracket. - Peter Luschny, Jan 24 2009

From Reikku Kulon, Feb 26 2009: (Start)

A109449(n, 0) = A000111(n), approx. round(2^(n + 2) * n! / Pi^(n + 1)).

A109449(n, n - 1) = n.

A109449(n, n) = 1.

For n > 0, k > 0: A109449(n, k) = A109449(n - 1, k - 1) * n / k. (End)

From Peter Luschny, Jul 10 2009: (Start)

Let p_n(x) = Sum_{k=0..n} Sum_{v=0..k} (-1)^v C(k,v)F(k)(x+v+1)^n, where F(0)=1 and for k>0 F(k)=-1 + s_k 2^floor((k-1)/2), s_k is 0 if k mod 8 in {2,6}, 1 if k mod 8 in {0,1,7} and otherwise -1. T(n,k) are the absolute values of the coefficients of these polynomials.

Another way to express the polynomials p_n(x) is

p_n(x) = -x^n + Sum_{k=0..n} binomial(n,k) Euler(k)((x+1)^(n-k)+x^(n-k)). (End)

From Peter Bala, Jan 26 2011: (Start)

An explicit formula for the n-th row polynomial is

x^n + i*Sum_{k=1..n}((1+i)/2)^(k-1)*Sum_{j=0..k} (-1)^j*binomial(k,j)*(x+i*j)^n, where i = sqrt(-1). This is the triangle of connection constants between the polynomial sequences {Z(n,x+1)} and {Z(n,x)}, where Z(n,x) denotes the zigzag polynomials described in A147309.

Denote the present array by M. The first column of the array (I-x*M)^-1 is a sequence of rational functions in x whose numerator polynomials are the row polynomials of A145876 - the generalized Eulerian numbers associated with the zigzag numbers. (End)

Let skp{n}(x) denote the Swiss-Knife polynomials A153641. Then

T(n,k) = [x^(n-k)] |skp{n}(x) - skp{n}(x-1) + x^n|. - Peter Luschny, Jul 22 2012

T(n,k) = A007318(n,k) * A000111(n - k), k = 0..n. - Reinhard Zumkeller, Nov 02 2013

T(n,k) = abs(A247453(n,k)). - Reinhard Zumkeller, Sep 17 2014

Triangle starts:

      1;

      1,     1;

      1,     2,     1;

      2,     3,     3,     1;

      5,     8,     6,     4,     1;

     16,    25,    20,    10,     5,    1;

     61,    96,    75,    40,    15,    6,    1;

    272,   427,   336,   175,    70,   21,    7,   1;

   1385,  2176,  1708,   896,   350,  112,   28,   8,  1;

   7936, 12465,  9792,  5124,  2016,  630,  168,  36,  9,  1;

  50521, 79360, 62325, 32640, 12810, 4032, 1050, 240, 45, 10, 1; ...

From Peter Luschny, Jul 10 2009: (Start)

# Auxiliary functions

Pow := (n, k) -> `if`(n=0 and k=0, 1, n^k): # To avoid '0^0 undefined'.

Euler := (n, x) -> `if`(n=0, 1, euler(n, x)): # Avoid the bug euler(0, 1) = -1.

sigma := proc(n) local nmod8; nmod8 := n mod 8;

if n = 0 then RETURN(1) fi; if member(nmod8, {2, 6}) then RETURN(-1) fi;

if member(nmod8, {0, 1, 7}) then 1 else -1 fi; %*2^(-iquo(n-1, 2))-1 end:

A000111 := n -> 2^n*abs(Euler(n, 1/2)+Euler(n, 1))-`if`(n=0, 1, 0):

# Coefficients

A109449 := proc(n, k) binomial(n, k)*A000111(n-k) end:

B109449 := proc(n, k) 2^(n-k)*binomial(n, k)*abs(Euler(n-k, 1/2)+Euler(n-k, 1)) -`if`(n-k=0, 1, 0) end:

R109449 := proc(n, k) option remember; if k = 0 then RETURN(A000111(n)) fi; R109449(n-1, k-1)*n/k end:

# Polynomials

E109449 := proc(n) local k; add(binomial(n, k)*euler(k)*(Pow(x+1, n-k)+ Pow(x, n-k)), k=0..n)-Pow(x, n) end:

L109449 := proc(n) local k, v; add(add((-1)^v*binomial(k, v)*Pow(x+v+1, n)* sigma(k), v=0..k), k=0..n) end:

X109449 := proc(n) n!*coeff(series(exp(x*t)*(sech(t)+tanh(t)), t, 24), t, n)end:

# Evaluate

seq(print(seq(A109449(n, k), k=0..n)), n=0..9);

seq(print(seq(B109449(n, k), k=0..n)), n=0..9);

seq(print(seq(R109449(n, k), k=0..n)), n=0..9);

seq(print(seq(abs(coeff(E109449(n), x, k)), k=0..n)), n=0..9);

seq(print(seq(abs(coeff(L109449(n), x, k)), k=0..n)), n=0..9);

seq(print(seq(abs(coeff(X109449(n), x, k)), k=0..n)), n=0..9); (End)

lim = 10; s = CoefficientList[Series[(1 + Sin[x])/Cos[x], {x, 0, lim}], x] Table[k!, {k, 0, lim}]; Table[Binomial[n, k] s[[n - k + 1]], {n, 0, lim}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 24 2015, after Jean-François Alcover at A000111 *)

T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 27 2019 *)

(Sage)

R = PolynomialRing(ZZ, 'x')

@CachedFunction

def skp(n, x) :

    if n == 0 : return 1

    return add(skp(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2])

def A109449_row(n):

    x = R.gen()

    return [abs(c) for c in list(skp(n, x)-skp(n, x-1)+x^n)]

for n in (0..10) : print(A109449_row(n)) # Peter Luschny, Jul 22 2012

(Haskell)

a109449 n k = a109449_row n !! k

a109449_row n = zipWith (*)

                (a007318_row n) (reverse $ take (n + 1) a000111_list)

a109449_tabl = map a109449_row [0..]

-- Reinhard Zumkeller, Nov 02 2013

(PARI) A109449(n, k)=binomial(n, k)*if(n>k, 2*abs(polylog(k-n, I)), 1) \\ M. F. Hasler, Oct 05 2017

Cf. A000111, A000667, A000795, A002084, A003719, A007318, A009747.

See also : A000182, A000964, A009739, A062161, A062272.

Cf. A153641, A162660.

Cf. A000667 (row sums), A247453.

Sequence in context: A238281 A080850 A247453 * A129570 A238385 A215652

Adjacent sequences:  A109446 A109447 A109448 * A109450 A109451 A109452

nonn,tabl

Philippe Deléham, Aug 27 2005

Edited, formula corrected, typo T(9,4)=2016 (before 2816) fixed by Peter Luschny, Jul 10 2009

approved