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%I #18 Feb 25 2021 08:32:23
%S 1,3,15,99,801,7623,83079,1017495,13808097,205374123,3318673599,
%T 57845821707,1081091446785,21553820597871,456410531639799,
%U 10225931132021247,241609515712343361,6002109578246918355,156360266121378584943,4261404847790207796147
%N Expansion of e.g.f.: exp(Sum_{k>=1} 3*x^k).
%C In general, if e.g.f. = exp(Sum_{k>=1} m*x^k) = exp(m*x/(1-x)) and m>0, then a(n) ~ n! * m^(1/4) * exp(2*sqrt(m*n) - m/2) / (2 * sqrt(Pi) * n^(3/4)).
%H G. C. Greubel, <a href="/A255806/b255806.txt">Table of n, a(n) for n = 0..435</a>
%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F E.g.f.: exp(3*x/(1-x)).
%F a(n) ~ 3^(1/4) * exp(2*sqrt(3*n) - 3/2) * n! / (2*sqrt(Pi)*n^(3/4)).
%F a(n) = (2*n+1)*a(n-1) - (n-2)*(n-1)*a(n-2). - _Vaclav Kotesovec_, Nov 04 2016
%F From _G. C. Greubel_, Feb 24 2021: (Start)
%F a(n) = A253286(n+3, 3).
%F a(n) = 3*(n-1)!*LaguerreL(n-1, 1, -3) with a(0) = 1. (End)
%t nmax=20; CoefficientList[Series[Exp[Sum[3*x^k,{k,1,nmax}]],{x,0,nmax}],x] * Range[0,nmax]!
%t CoefficientList[Series[E^(3*x/(1-x)), {x, 0, 20}], x] * Range[0, 20]!
%t Table[If[n==0, 1, 3*(n-1)!*LaguerreL[n-1, 1, -3]], {n, 0, 25}] (* _G. C. Greubel_, Feb 24 2021 *)
%o (PARI) my(x='x +O('x^50)); Vec(serlaplace(exp(3*x/(1-x)))) \\ _G. C. Greubel_, Feb 05 2017
%o (Sage) [1 if n==0 else 3*factorial(n-1)*gen_laguerre(n-1, 1, -3) for n in (0..25)] # _G. C. Greubel_, Feb 24 2021
%o (Magma) [n eq 0 select 1 else 3*Factorial(n-1)*Evaluate(LaguerrePolynomial(n-1, 1), -3): n in [0..25]]; // _G. C. Greubel_, Feb 24 2021
%Y Cf. A000262, A052897, A253286.
%K nonn,easy
%O 0,2
%A _Vaclav Kotesovec_, Mar 07 2015
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