A007113 Expansion of e.g.f. (1 + x)^x. (Formerly M0919)

1, 0, 2, -3, 20, -90, 594, -4200, 34544, -316008, 3207240, -35699400, 432690312, -5672581200, 79991160144, -1207367605080, 19423062612480, -331770360922560, 5997105160795584, -114373526841360000, 2295170834453089920, -48344592370577247360

Offset

0,3

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.3.

Links

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 75

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*k!*Stirling1(n-k, k). - Vladeta Jovovic, Dec 19 2004

a(n) ~ (-1)^n * n!. - Vaclav Kotesovec, Jun 06 2019

Maple

a:= n-> n! *coeff(series((1+x)^x, x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 12 2012

Mathematica

CoefficientList[Series[(1 + x)^x, {x, 0, 19}], x]*Table[(n - 1)!, {n, 1, 20}]
a[n_] := (-1)^n*n!*Sum[ StirlingS1[n - k, k]/(n - k)!*(-1)^(n - 2*k), {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Dec 12 2012, after Vladeta Jovovic *)

Crossrefs

Cf. A053489, A053490. Apart from initial terms and signs, same as A066166.

Sequence in context: A264417 A348311 A066166 * A052804 A267652 A258089

Adjacent sequences: A007110 A007111 A007112 * A007114 A007115 A007116

Keyword

sign

Author

Simon Plouffe

Extensions

Signs from Christian G. Bower, Nov 15 1998

Status

approved