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%I #11 Nov 16 2022 05:21:37
%S 1,0,-2,0,-3,3,0,-4,12,0,0,-5,35,0,-30,0,-6,90,0,-360,180,0,-7,217,0,
%T -2730,3150,-630,0,-8,504,0,-16800,33600,-15120,0,0,-9,1143,0,-91854,
%U 283500,-215460,0,22680,0,-10,2550,0,-466200,2085300,-2381400,0,907200,-226800
%N Triangle read by rows. G(n, k) an additive decomposition of 2^n*G(n), G(n) the Genocchi numbers.
%C The Swiss-Knife polynomials A153641 can be understood as a sum of polynomials. Evaluated at x=-1 multiplied by n+1 this results in a decomposition of 2^n times the Genocchi numbers A036968.
%H G. C. Greubel, <a href="/A154344/b154344.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 G(n,k) = Sum_{v=0,..,k} ( (-1)^(v)*binomial(k,v)*(n+1)*c(k)*v^n );
%F G(n) = (1/2^n)*Sum_{k=0,..,n} G(n,k).
%e 1,
%e 0, -2,
%e 0, -3, 3,
%e 0, -4, 12, 0,
%e 0, -5, 35, 0, -30,
%e 0, -6, 90, 0, -360, 180,
%e 0, -7, 217, 0, -2730, 3150, -630,
%e 0, -8, 504, 0, -16800, 33600, -15120, 0,
%e 0, -9, 1143, 0, -91854, 283500, -215460, 0, 22680.
%p G := 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)*(n+1)*c(k)*v^n, v=0..k) end: seq(print(seq(G(n, k), k=0..n)), n=0..8);
%t g[n_, k_] := Module[{v, c, pow}, pow[a_, b_] := If[ a == 0 && b == 0, 1, a^b]; c[m_] := If[ Mod[m+1, 4] == 0 , 0 , 1/((-1)^Quotient[m+1, 4]*2^Quotient[m, 2])]; Sum[(-1)^v*Binomial[k, v]*(n+1)*c[k]*pow[v, n], {v, 0, k}]]; Table[g[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 23 2013, translated from Maple *)
%Y Cf. A153641, A154341, A154342, A154343, A154345.
%K easy,sign,tabl
%O 0,3
%A _Peter Luschny_, Jan 07 2009
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