A081046 Difference of the first two Stirling numbers of the first kind.

1, -2, 5, -17, 74, -394, 2484, -18108, 149904, -1389456, 14257440, -160460640, 1965444480, -26029779840, 370643938560, -5646837369600, 91657072281600, -1579093018675200, 28779361764249600, -553210247226470400

Offset: 0

Author

Paul Barry, Mar 05 2003

Keywords

• easy
• sign

Crossrefs

Cf. A000254, A008275. Same as A000774 apart from signs.

Programs

Mathematica

Table[StirlingS1[n+1, 1] - StirlingS1[n+1, 2], {n, 0, 20}] (* or *) Table[(-1)^n n! (1+HarmonicNumber[n]), {n, 0, 20}] (* Jean-François Alcover, Feb 11 2016 *)

PARI

a(n) = stirling(n+1, 1, 1) - stirling(n+1, 2, 1); \\ Michel Marcus, Feb 11 2016

Formula

a(n) = s(n, 1)-s(n, 2), s(n, m) = signed Stirling number of the first kind.

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

Conjecture: a(n) +(2*n-1)*a(n-1) +(n-1)^2*a(n-2)=0. - R. J. Mathar, Oct 27 2014