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Expansion of e.g.f.: (1-x)^(-1/2)*exp(-x/2 -x^2/4 -x^3/6 -x^4/8).
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%I #10 Sep 08 2022 08:45:00

%S 1,0,0,0,0,12,60,360,2520,20160,199584,2147040,25043040,315485280,

%T 4274281440,62237343168,968728662720,16046598597120,281802435747840,

%U 5229395457937920,102253297006250496,2101387824575550720,45281611027331723520

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

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

%H G. C. Greubel, <a href="/A053533/b053533.txt">Table of n, a(n) for n = 0..445</a>

%F a(n) ~ sqrt(2) * n^n / exp(n+25/24). - _Vaclav Kotesovec_, Aug 04 2014

%t 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 *)

%o (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

%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 -x^4/8) )); [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^4/8), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, May 15 2019

%Y Cf. A001205, A053532.

%K nonn

%O 0,6

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