A053532 Expansion of e.g.f.: (1-x)^(-1/2)*exp(-x/2 -x^2/4 -x^3/6).

1, 0, 0, 0, 3, 12, 60, 360, 2835, 24696, 237384, 2503440, 28941165, 363593340, 4930388892, 71759200968, 1115892704745, 18465120087120, 323965034820720, 6007037150742624, 117377605956803571, 2410702829834021820, 51917379915449131020

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=4.

Links

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

Crossrefs

Cf. A001205, A053533.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1-x)^(-1/2)*Exp(-x/2 -x^2/4 -x^3/6) )); [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, 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) )) \\ 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, 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+11/12). - Vaclav Kotesovec, Aug 04 2014

Conjecture: D-finite with recurrence 2*a(n) +2*(-n+1)*a(n-1) -(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Jul 06 2020