A053533 Expansion of e.g.f.: (1-x)^(-1/2)*exp(-x/2 -x^2/4 -x^3/6 -x^4/8).

1, 0, 0, 0, 0, 12, 60, 360, 2520, 20160, 199584, 2147040, 25043040, 315485280, 4274281440, 62237343168, 968728662720, 16046598597120, 281802435747840, 5229395457937920, 102253297006250496, 2101387824575550720, 45281611027331723520

Offset: 0

Author

N. J. A. Sloane, Jan 16 2000

Keywords

• nonn

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.15(a), k=5.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..445

Crossrefs

Cf. A001205, A053532.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1-x)^(-1/2)*Exp(-x/2 -x^2/4 -x^3/6 -x^4/8) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019

Mathematica

With[{m = 30}, CoefficientList[Series[(1-x)^(-1/2)*Exp[-x/2 -x^2/4 -x^3/6 -x^4/8], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 15 2019 *)

PARI

my(x='x+O('x^30)); Vec(serlaplace( (1-x)^(-1/2)*exp(-x/2 -x^2/4 -x^3/6 -x^4/8) )) \\ G. C. Greubel, May 15 2019

Sage

m = 30; T = taylor((1-x)^(-1/2)*exp(-x/2 -x^2/4 -x^3/6 -x^4/8), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019

Formula

a(n) ~ sqrt(2) * n^n / exp(n+25/24). - Vaclav Kotesovec, Aug 04 2014