A081048 Signed Stirling numbers of the first kind.

0, 1, -3, 11, -50, 274, -1764, 13068, -109584, 1026576, -10628640, 120543840, -1486442880, 19802759040, -283465647360, 4339163001600, -70734282393600, 1223405590579200, -22376988058521600, 431565146817638400, -8752948036761600000, 186244810780170240000

Offset: 0

Author

Paul Barry, Mar 05 2003

Keywords

• sign

Examples

			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
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Vladimir Reshetnikov, Proof Mathar's formula, Apr 24 2013

Crossrefs

Cf. A000254, A008275.

Programs

Magma

m:=30; R:=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

Maple

a:= proc(n) option remember;
      `if`(n<2, n, (1-2*n)*a(n-1) -(n-1)^2*a(n-2))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 06 2013

Mathematica

aa = {}; Do[AppendTo[aa,Coefficient[Expand[Product[p - n, {n, 0, m}]], p, 2]], {m, 1, 20}]; aa (* Artur Jasinski, Nov 30 2008 *)
a[n_] := (-1)^(n+1)*n!*HarmonicNumber[n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 29 2017 *)
Table[StirlingS1[n, 2], {n, 1, 20}] (* Vaclav Kotesovec, Mar 03 2022 *)

PARI

a(n)=stirling(n,2) \\ Charles R Greathouse IV, May 08 2015

Formula

a(n) = n!*Sum {k=1..n} (-1)^(n+1)*1/k.

E.g.f.: log(1+x)/(1+x).

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

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

a(n) ~ n! * (-1)^(n+1) * (log(n) + gamma), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 05 2013