A353253 Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (j - x).

1, 0, -1, -1, -1, -3, -14, -76, -480, -3491, -28792, -265708, -2713753, -30395515, -370509784, -4883351213, -69205187838, -1049436525897, -16956113955333, -290817728309779, -5277059794403117, -101005287980087110, -2033813167589257170, -42977173319758429942

Offset: 0

Author

Seiichi Manyama, Apr 08 2022

Keywords

• sign

Crossrefs

Cf. A343579, A353252, A353254.

Cf. A353260.

Programs

Mathematica

a[n_] := Sum[(-1)^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-x)))

PARI

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

Formula

a(n) = Sum_{k=0..floor(n/2)} (-1)^k * |Stirling1(n-k,k)|.