0,3

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.

Peter Luschny, The Swiss-Knife polynomials.

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) = (Sum(k=0..n, G(n,k)) / 2^n

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,

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);

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

Sequence in context: A138057 A053727 A127952 this_sequence A134409 A094067 A094112

Adjacent sequences: A154341 A154342 A154343 this_sequence A154345 A154346 A154347

easy,sign,tabl

Peter Luschny (peter(AT)luschny.de), Jan 07 2009