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Expansion of e.g.f.: (1-x)^(-1/2)*exp(-x/2 -x^2/4 -x^3/6).
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%I #14 Sep 08 2022 08:45:00

%S 1,0,0,0,3,12,60,360,2835,24696,237384,2503440,28941165,363593340,

%T 4930388892,71759200968,1115892704745,18465120087120,323965034820720,

%U 6007037150742624,117377605956803571,2410702829834021820,51917379915449131020

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

%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.15(a), k=4.

%H G. C. Greubel, <a href="/A053532/b053532.txt">Table of n, a(n) for n = 0..445</a>

%F a(n) ~ sqrt(2) * n^n / exp(n+11/12). - _Vaclav Kotesovec_, Aug 04 2014

%F 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

%t 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 *)

%o (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

%o (Magma) m:=30; R<x>:=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

%o (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

%Y Cf. A001205, A053533.

%K nonn

%O 0,5

%A _N. J. A. Sloane_, Jan 16 2000