0,5

The boustrophedon transform {t} of a sequence {s} is given by t_n = Sum_{k, 0<=k<=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]. [From Peter Luschny (peter(AT)luschny.de), Jan 24 2009]

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

Peter Luschny, The Swiss-Knife polynomials. [From Peter Luschny (peter(AT)luschny. de), Jul 10 2009]

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).

E.g.f.: exp(x*y)*(sec(x)+tan(x)). - Vladeta Jovovic (vladeta(AT)eunet.rs), 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. [From Peter Luschny (peter(AT)luschny.de), Jan 24 2009]

Contribution from Reikku Kulon (reikku(AT)gmail.com), 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)

Contribution from Peter Luschny (peter(AT)luschny.de), 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} C(n,k) Euler(k)((x+1)^(n-k)+x^(n-k)). (End)

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; ...

Contribution from Peter Luschny (peter(AT)luschny.de), 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)

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

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

Cf. A153641, A162660. [From Peter Luschny (peter(AT)luschny.de), Jul 10 2009]

Sequence in context: A097724 A091836 A080850 this_sequence A129570 A165014 A058063

Adjacent sequences: A109446 A109447 A109448 this_sequence A109450 A109451 A109452

nonn,tabl

Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 27 2005

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