A081051 Stirling numbers of the first kind.

0, 0, 1, -6, 35, -225, 1624, -13132, 118124, -1172700, 12753576, -150917976, 1931559552, -26596717056, 392156797824, -6165817614720, 102992244837120, -1821602444624640, 34012249593822720, -668609730341153280, 13803759753640704000, -298631902863216384000

Offset: 0

Author

Paul Barry, Mar 05 2003

Keywords

• easy
• sign

Comments

Coefficient of x^3 in Product {k=0..(n-1), x-k}.

Crossrefs

Cf. A000399, A008275, A081048.

Programs

Mathematica

Table[StirlingS1[n, 3], {n, 1, 20}] (* Vaclav Kotesovec, Mar 03 2022 *)

Formula

E.g.f. (1+x)^(-1)*log(1+x)^2/2

a(n) = (-1)^n*det(S(i+3,j+2), 1 <= i,j <= n-2), where S(n,k) are Stirling numbers of the second kind and n>1. [Mircea Merca, Apr 06 2013]

a(n) ~ n! * (-1)^n * log(n)^2/2 * (1 + 2*gamma/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 03 2022