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A081048 Signed Stirling numbers of the first kind. 5
0, 1, -3, 11, -50, 274, -1764, 13068, -109584, 1026576, -10628640, 120543840, -1486442880, 19802759040, -283465647360, 4339163001600, -70734282393600, 1223405590579200, -22376988058521600, 431565146817638400, -8752948036761600000, 186244810780170240000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Vladimir Reshetnikov, Proof Mathar's formula, Apr 24 2013

FORMULA

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

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

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

EXAMPLE

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

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-Fran├žois Alcover, Mar 29 2017 *)

PROG

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

CROSSREFS

Cf. A000254, A008275.

Sequence in context: A230961 A203166 A000254 * A065048 A256126 A024335

Adjacent sequences:  A081045 A081046 A081047 * A081049 A081050 A081051

KEYWORD

sign

AUTHOR

Paul Barry, Mar 05 2003

STATUS

approved

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Last modified November 19 07:02 EST 2017. Contains 294915 sequences.