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A081048 Signed Stirling numbers of the first kind.

%I A081048 #43 Sep 08 2022 08:45:09
%S A081048 0,1,-3,11,-50,274,-1764,13068,-109584,1026576,-10628640,120543840,
%T A081048 -1486442880,19802759040,-283465647360,4339163001600,-70734282393600,
%U A081048 1223405590579200,-22376988058521600,431565146817638400,-8752948036761600000,186244810780170240000
%N A081048 Signed Stirling numbers of the first kind.
%H A081048 Vincenzo Librandi, <a href="/A081048/b081048.txt">Table of n, a(n) for n = 0..200</a>
%H A081048 Vladimir Reshetnikov, <a href="/A081048/a081048.txt">Proof Mathar's formula, Apr 24 2013</a>
%F A081048 a(n) = n!*Sum {k=1..n} (-1)^(n+1)*1/k.
%F A081048 E.g.f.: log(1+x)/(1+x).
%F A081048 D-finite with recurrence a(n) = (2*n-1)*a(n-1) + (n-1)^2*a(n-2) = 0. (Proved by Reshetnikov.) - _R. J. Mathar_, Nov 24 2012
%F A081048 a(n) = (-1)^(n-1)*det(S(i+2,j+1), 1 <= i,j <= n-1), where S(n,k) are Stirling numbers of the second kind and n>0. - _Mircea Merca_, Apr 06 2013
%F A081048 a(n) ~ n! * (-1)^(n+1) * (log(n) + gamma), where gamma is the Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Oct 05 2013
%e A081048 a(9): coefficient of p^2 in polynomial p (p - 1) (p - 2) (p - 3) (p - 4) (p - 5) (p - 6) (p - 7) (p - 8) = -1 + 40320 p - 109584 p^2 + 118124 p^3 - 67284 p^4 + 22449 p^5 - 4536 p^6 + 546 p^7 - 36 p^8 + p^9 is equal to -109584. - _Artur Jasinski_, Nov 30 2008
%p A081048 a:= proc(n) option remember;
%p A081048       `if`(n<2, n, (1-2*n)*a(n-1) -(n-1)^2*a(n-2))
%p A081048     end:
%p A081048 seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 06 2013
%t A081048 aa = {}; Do[AppendTo[aa,Coefficient[Expand[Product[p - n, {n, 0, m}]], p, 2]], {m, 1, 20}]; aa (* _Artur Jasinski_, Nov 30 2008 *)
%t A081048 a[n_] := (-1)^(n+1)*n!*HarmonicNumber[n];
%t A081048 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 29 2017 *)
%t A081048 Table[StirlingS1[n, 2], {n, 1, 20}] (* _Vaclav Kotesovec_, Mar 03 2022 *)
%o A081048 (PARI) a(n)=stirling(n,2) \\ _Charles R Greathouse IV_, May 08 2015
%o A081048 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Log(1+x)/(1+x))); [0] cat [Factorial(n)*b[n]: n in [1..m-1]]; // _G. C. Greubel_, Aug 28 2018
%Y A081048 Cf. A000254, A008275.
%K A081048 sign
%O A081048 0,3
%A A081048 _Paul Barry_, Mar 05 2003