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%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
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