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Stirling numbers of the first kind.
4

%I #9 Mar 03 2022 06:10:59

%S 0,0,1,-6,35,-225,1624,-13132,118124,-1172700,12753576,-150917976,

%T 1931559552,-26596717056,392156797824,-6165817614720,102992244837120,

%U -1821602444624640,34012249593822720,-668609730341153280,13803759753640704000,-298631902863216384000

%N Stirling numbers of the first kind.

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

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

%F 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]

%F 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

%t Table[StirlingS1[n, 3], {n, 1, 20}] (* _Vaclav Kotesovec_, Mar 03 2022 *)

%Y Cf. A000399, A008275, A081048.

%K easy,sign

%O 0,4

%A _Paul Barry_, Mar 05 2003