0,5

The Swiss-Knife polynomials A153641 can be understood as a sum of polynomials. Evaluated at x=0 these polynomials result in a decomposition of the Euler number A122045.

Peter Luschny, The Swiss-Knife polynomials.

Let c(k) = frac{(-1)^{floor(k/4)}{2^{floor(k/2)}} [4 not div k] (Iverson notation).

E(n,k) = Sum(v=0..k, (-1)^(v)*binomial(k,v)*c(k)*(v+1)^n);

E(n) = Sum(k=0..n, E(n,k)).

1,

1, -1,

1, -3, 1,

1, -7, 6, 0,

1, -15, 25, 0, -6,

1, -31, 90, 0, -90, 30,

1, -63, 301, 0, -840, 630, -90,

1, -127, 966, 0, -6300, 7980, -2520, 0,

1, -255, 3025, 0, -41706, 79380, -41580, 0, 2520,

E := 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)*c(k)*(v+1)^n, v=0..k) end: seq(print(seq(E(n, k), k=0..n)), n=0..8);

Cf. A153641,A154342,A154343,A154344,A154345.

Sequence in context: A080936 A094507 A065625 this_sequence A130749 A154959 A008277

Adjacent sequences: A154338 A154339 A154340 this_sequence A154342 A154343 A154344

easy,sign,tabl

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