A007114 Expansion of e.g.f. (1 - x)^x.

1, 0, -2, -3, 4, 30, 66, 0, -496, -1512, 1800, 51480, 487752, 4633200, 50605296, 620703720, 8278947840, 118504008000, 1811156124096, 29452505385600, 507926275873920, 9260774050469760, 178004846331607680, 3597786477984061440, 76284908749415574144, 1693206585883612800000, 39264054809925638534400

Offset: 0

Author

Simon Plouffe

Keywords

• sign

Comments

It appears that a(n) is always positive for n > 9.

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Branko Dragovich, On Summation of p-Adic Series, arXiv:1702.02569 [math.NT], 2017.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1-x)^x)); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 30 2018

Mathematica

CoefficientList[Series[(1-x)^x,{x,0,30}],x]*Table[(n-1)!,{n,1,31}] (* Vincenzo Librandi, Jun 16 2012 *)

PARI

a(n) = sum(k=0,n, (-1)^k*stirling(k,n-k)*n!/k!) \\ Max Alekseyev, Aug 10 2013

PARI

x='x+O('x^33); Vec(serlaplace(exp(x*log(1 - x)))) \\ Joerg Arndt, Apr 01 2017

Formula

a(n) = Sum_{k=0..n} (-1)^k * A008275(k,n-k) n!/k!. - Max Alekseyev, Aug 10 2013

a(n) ~ 2 * n! / n^3 * (1 + (17/2-3*log(n)-3*gamma)/n), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Apr 21 2014

Extensions

Signs from Christian G. Bower, Nov 15 1998