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A154344
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Triangle read by rows. G(n, k) an additive decomposition of 2^n*G(n), G(n) the Genocchi numbers.
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6
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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, -2730, 3150, -630, 0, -8, 504, 0, -16800, 33600, -15120, 0, 0, -9, 1143, 0, -91854, 283500, -215460, 0, 22680, 0, -10, 2550, 0, -466200, 2085300, -2381400, 0, 907200, -226800
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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Let c(k) = frac{(-1)^{floor(k/4)}{2^{floor(k/2)}} [4 not div k] (Iverson notation).
G(n,k) = Sum_{v=0,..,k} ( (-1)^(v)*binomial(k,v)*(n+1)*c(k)*v^n );
G(n) = (1/2^n)*Sum_{k=0,..,n} G(n,k).
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EXAMPLE
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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, -2730, 3150, -630,
0, -8, 504, 0, -16800, 33600, -15120, 0,
0, -9, 1143, 0, -91854, 283500, -215460, 0, 22680.
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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