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%I #19 Sep 08 2022 08:45:12
%S 1,5,49,683,12181,263093,6650245,192153587,6238115689,224551351493,
%T 8869372524409,381149538287675,17695559832649021,882309688871504117,
%U 47006884504348992589,2664275436650330250947,160032535163170368513745,10152680666397062173751813,678253866638401212881394241
%N a(n) = 3^n *n! *L_{n}^{-1/3}(-1), where L_n^{alpha}(x) are generalized Laguerre polynomials.
%H G. C. Greubel, <a href="/A089914/b089914.txt">Table of n, a(n) for n = 0..375</a>
%F a(n) ~ n^(n+1/12)*3^n*exp(-n+2*sqrt(n)-1/2)/sqrt(2) * (1 + 65/(144*sqrt(n))). - _Vaclav Kotesovec_, Jun 22 2013
%F From _Peter Bala_, Jun 14 2014: (Start)
%F E.g.f.: 1/(1 - 3*x)^(2/3)*exp(3*x/(1 - 3*x)) = 1 + 5*x + 49*x^2/2! + 683*x^3/3! + ....
%F Dobinski-type formula: a(n) = (3^n/exp(1))*Sum {k >= 0} (n!/k!)* binomial(n + k - 1/3,k - 1/3).
%F Recurrence equation: a(n) = (6*n - 1)a(n-1) - (n - 1)*(9*n - 12)*a(n-2) with a(0) = 1 and a(1) = 5. (End)
%p A089914 := proc(n)
%p 3^n*n!*LaguerreL(n,-1/3,-1) ;
%p simplify(%) ;
%p end proc;
%t Table[3^n*n!*LaguerreL[n,-1/3,-1],{n,0,20}] (* _Vaclav Kotesovec_, Jun 22 2013 *)
%o (PARI) x='x+O('x^30); Vec(serlaplace(1/(1 - 3*x)^(2/3)*exp(3*x/(1 - 3*x)))) \\ _G. C. Greubel_, May 13 2018
%o (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients( R!(1/(1 - 3*x)^(2/3)*Exp(3*x/(1 - 3*x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 13 2018
%K nonn
%O 0,2
%A _Karol A. Penson_, Nov 14 2003
%E Terms a(15) onward added by _G. C. Greubel_, May 13 2018
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