A154343 S(n,k) an additive decomposition of the Springer number (generalized Euler number), (triangle read by rows).

1, 3, -2, 9, -16, 4, 27, -98, 60, 0, 81, -544, 616, 0, -96, 243, -2882, 5400, 0, -3360, 960, 729, -14896, 43564, 0, -72480, 46080, -5760, 2187, -75938, 334740, 0, -1246560, 1323840, -362880, 0, 6561, -384064, 2495056, 0, -18801216, 29675520

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=1/2 and multiplied by 2^n these polynomials result in a decomposition of the Springer numbers A001586.

Examples

			Triangle begins:
  1,
  3,    -2,
  9,    -16,     4,
  27,   -98,     60,      0,
  81,   -544,    616,     0, -96,
  243,  -2882,   5400,    0, -3360,     960,
  729,  -14896,  43564,   0, -72480,    46080,    -5760,
  2187, -75938,  334740,  0, -1246560,  1323840,  -362880,   0,
  6561, -384064, 2495056, 0, -18801216, 29675520, -13386240, 0, 645120,
  ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1274
• Peter Luschny, The Swiss-Knife polynomials.

Crossrefs

Cf. A001586, A188458.

Cf. A153641, A154341, A154342, A154344, A154345.

Programs

Maple

S := 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)*2^n*c(k)*(v+3/2)^n,v=0..k) end: seq(print(seq(S(n,k),k=0..n)),n=0..8);

Mathematica

c[m_] := If[Mod[m+1, 4] == 0, 0, 1/((-1)^Quotient[m+1, 4]*2^Quotient[m, 2])]; s[n_, k_] := Sum[(-1)^v*Binomial[k, v]*2^n*c[k]*(v+3/2)^n, {v, 0, k}]; Table[s[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2013, after Maple *)

Formula

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

S(n,k) = Sum_{v=0..k} (-1)^v*binomial(k,v)*2^n*c(k)*(v+3/2)^n.

A188458(n) = Sum_{k=0..n} S(n,k).