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A154344 G(n,k) an additive decomposition of 2^n*G(n), G(n) the Genocchi numbers (triangle read by rows). 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1274

Peter Luschny, The Swiss-Knife polynomials.

FORMULA

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).

EXAMPLE

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.

MAPLE

G := proc(n, k) local v, c, pow; pow := (a, b) -> if a = 0 and b = 0 then 1 else a^b fi; 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)*pow(v, n), v=0..k) end: seq(print(seq(G(n, k), k=0..n)), n=0..8);

MATHEMATICA

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-Fran├žois Alcover, May 23 2013, translated from Maple *)

CROSSREFS

Cf. A153641,A154341,A154342,A154343,A154345.

Sequence in context: A127952 A171307 A209693 * A134409 A180013 A094067

Adjacent sequences:  A154341 A154342 A154343 * A154345 A154346 A154347

KEYWORD

easy,sign,tabl

AUTHOR

Peter Luschny, Jan 07 2009

STATUS

approved

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Last modified September 22 09:48 EDT 2017. Contains 292337 sequences.