A154343 - OEIS

%I #15 Sep 13 2016 02:54:01

%S 1,3,-2,9,-16,4,27,-98,60,0,81,-544,616,0,-96,243,-2882,5400,0,-3360,

%T 960,729,-14896,43564,0,-72480,46080,-5760,2187,-75938,334740,0,

%U -1246560,1323840,-362880,0,6561,-384064,2495056,0,-18801216,29675520

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

%C 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.

%H G. C. Greubel, <a href="/A154343/b154343.txt">Table of n, a(n) for n = 0..1274</a>

%H Peter Luschny, <a href="http://www.luschny.de/math/seq/SwissKnifePolynomials.html">The Swiss-Knife polynomials.</a>

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

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

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

%e 1,

%e 3,     -2,

%e 9,     -16,      4,

%e 27,    -98,     60,      0,

%e 81,    -544,    616,     0,   -96,

%e 243,   -2882,   5400,    0,  -3360,      960,

%e 729,   -14896,  43564,   0,  -72480,    46080,     -5760,

%e 2187,  -75938,  334740,  0, -1246560,  1323840,   -362880,  0,

%e 6561, -384064,  2495056, 0, -18801216, 29675520, -13386240, 0, 645120.

%p 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);

%t 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 *)

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

%K easy,sign,tabl

%O 0,2

%A _Peter Luschny_, Jan 07 2009