A353254 Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (j - 2 * x).

1, 0, -2, -2, 0, 0, -12, -88, -608, -4664, -40032, -381200, -3993520, -45685472, -566975456, -7589393568, -109019255360, -1673050977024, -27321358963904, -473094230383616, -8659054324278528, -167044915214322816, -3387793305708038400, -72061754672510128384

Offset: 0

Author

Seiichi Manyama, Apr 08 2022

Keywords

• sign

Crossrefs

Cf. A343579, A353252, A353253.

Cf. A353261.

Programs

Mathematica

a[n_] := Sum[(-2)^k * Abs[StirlingS1[n - k, k]], {k, 0, Floor[n/2]}]; Array[a, 25, 0] (* Amiram Eldar, Apr 09 2022 *)

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, j-2*x)))

PARI

a(n) = sum(k=0, n\2, (-2)^k*abs(stirling(n-k, k, 1)));

Formula

a(n) = Sum_{k=0..floor(n/2)} (-2)^k * |Stirling1(n-k,k)|.