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

0,2

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.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1274

Peter Luschny, The Swiss-Knife polynomials.

Formula

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

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

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A076584 A309673 A378931 * A049969 A088634 A118791

Adjacent sequences: A154340 A154341 A154342 * A154344 A154345 A154346

Keyword

easy,sign,tabl

Author

Peter Luschny, Jan 07 2009

Status

approved