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

%I A154343 #18 Feb 03 2025 01:54:41
%S A154343 1,3,-2,9,-16,4,27,-98,60,0,81,-544,616,0,-96,243,-2882,5400,0,-3360,
%T A154343 960,729,-14896,43564,0,-72480,46080,-5760,2187,-75938,334740,0,
%U A154343 -1246560,1323840,-362880,0,6561,-384064,2495056,0,-18801216,29675520
%N A154343 S(n,k) an additive decomposition of the Springer number (generalized Euler number), (triangle read by rows).
%C A154343 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 A154343 G. C. Greubel, <a href="/A154343/b154343.txt">Table of n, a(n) for n = 0..1274</a>
%H A154343 Peter Luschny, <a href="http://www.luschny.de/math/seq/SwissKnifePolynomials.html">The Swiss-Knife polynomials.</a>
%F A154343 Let c(k) = ((-1)^floor(k/4) / 2^floor(k/2)) * [4 not div k+1] (Iverson notation).
%F A154343 S(n,k) = Sum_{v=0..k} (-1)^v*binomial(k,v)*2^n*c(k)*(v+3/2)^n.
%F A154343 A188458(n) = Sum_{k=0..n} S(n,k).
%e A154343 Triangle begins:
%e A154343   1,
%e A154343   3,    -2,
%e A154343   9,    -16,     4,
%e A154343   27,   -98,     60,      0,
%e A154343   81,   -544,    616,     0, -96,
%e A154343   243,  -2882,   5400,    0, -3360,     960,
%e A154343   729,  -14896,  43564,   0, -72480,    46080,    -5760,
%e A154343   2187, -75938,  334740,  0, -1246560,  1323840,  -362880,   0,
%e A154343   6561, -384064, 2495056, 0, -18801216, 29675520, -13386240, 0, 645120,
%e A154343   ...
%p A154343 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 A154343 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 A154343 Cf. A001586, A188458.
%Y A154343 Cf. A153641, A154341, A154342, A154344, A154345.
%K A154343 easy,sign,tabl
%O A154343 0,2
%A A154343 _Peter Luschny_, Jan 07 2009