A154344 Triangle read by rows. G(n, k) an additive decomposition of 2^n*G(n), G(n) the Genocchi numbers.

1, 0, -2, 0, -3, 3, 0, -4, 12, 0, 0, -5, 35, 0, -30, 0, -6, 90, 0, -360, 180, 0, -7, 217, 0, -2730, 3150, -630, 0, -8, 504, 0, -16800, 33600, -15120, 0, 0, -9, 1143, 0, -91854, 283500, -215460, 0, 22680, 0, -10, 2550, 0, -466200, 2085300, -2381400, 0, 907200, -226800

Offset: 0

Author

Peter Luschny, Jan 07 2009

Keywords

• easy
• sign
• tabl

Comments

The Swiss-Knife polynomials A153641 can be understood as a sum of polynomials. Evaluated at x=-1 multiplied by n+1 this results in a decomposition of 2^n times the Genocchi numbers A036968.

Examples

			Triangle begins:
  1,
  0, -2,
  0, -3,    3,
  0, -4,   12, 0,
  0, -5,   35, 0,    -30,
  0, -6,   90, 0,   -360,    180,
  0, -7,  217, 0,  -2730,   3150,    -630,
  0, -8,  504, 0, -16800,  33600,  -15120, 0,
  0, -9, 1143, 0, -91854, 283500, -215460, 0, 22680,
  ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1274
• Ayse Yilmaz Ceylan and Yilmaz Simsek, Formulae for Generalization of Touchard Polynomials with Their Generating Functions, Symmetry (2025) Vol. 17, Issue 7, Art. No. 1126. See Eq. 28 and after.
• Peter Luschny, The Swiss-Knife polynomials.

Crossrefs

Cf. A036968, A153641, A154341, A154342, A154343, A154345.

Programs

Maple

G := proc(n, k) local v, c; c := m -> if irem(m+1, 4) = 0 then 0 else 1/((-1)^iquo(m+1, 4)*2^iquo(m, 2)) fi; add((-1)^v*binomial(k, v)*(n+1)*c(k)*v^n, v=0..k) end: seq(print(seq(G(n, k), k=0..n)), n=0..8);

Mathematica

g[n_, k_] := Module[{v, c, pow}, pow[a_, b_] := If[ a == 0 && b == 0, 1, a^b]; c[m_] := If[ Mod[m+1, 4] == 0 , 0 , 1/((-1)^Quotient[m+1, 4]*2^Quotient[m, 2])]; Sum[(-1)^v*Binomial[k, v]*(n+1)*c[k]*pow[v, n], {v, 0, k}]]; Table[g[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 23 2013, translated from Maple *)

Formula

Let c(k) = ((-1)^floor(k/4) / 2^floor(k/2)) * [4 not div k+1] (Iverson notation).

G(n,k) = Sum_{v=0..k} (-1)^v*binomial(k,v)*(n+1)*c(k)*v^n.

A036968(n) = (1/2^n)*Sum_{k=0..n} G(n,k).