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A255806 Expansion of e.g.f.: exp(Sum_{k>=1} 3*x^k).

%I A255806 #20 Aug 24 2025 07:11:10
%S A255806 1,3,15,99,801,7623,83079,1017495,13808097,205374123,3318673599,
%T A255806 57845821707,1081091446785,21553820597871,456410531639799,
%U A255806 10225931132021247,241609515712343361,6002109578246918355,156360266121378584943,4261404847790207796147
%N A255806 Expansion of e.g.f.: exp(Sum_{k>=1} 3*x^k).
%C A255806 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 A255806 G. C. Greubel, <a href="/A255806/b255806.txt">Table of n, a(n) for n = 0..435</a>
%H A255806 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A255806 E.g.f.: exp(3*x/(1-x)).
%F A255806 a(n) ~ 3^(1/4) * exp(2*sqrt(3*n) - 3/2) * n! / (2*sqrt(Pi)*n^(3/4)).
%F A255806 a(n) = (2*n+1)*a(n-1) - (n-2)*(n-1)*a(n-2). - _Vaclav Kotesovec_, Nov 04 2016
%F A255806 From _G. C. Greubel_, Feb 24 2021: (Start)
%F A255806 a(n) = A253286(n+3, 3).
%F A255806 a(n) = 3*(n-1)!*LaguerreL(n-1, 1, -3) with a(0) = 1. (End)
%F A255806 For n > 0, a(n) = (n-1)! * Sum_{k=1..n} binomial(n,k) * 3^k / (k-1)!. - _Vaclav Kotesovec_, Aug 24 2025
%t A255806 nmax=20; CoefficientList[Series[Exp[Sum[3*x^k,{k,1,nmax}]],{x,0,nmax}],x] * Range[0,nmax]!
%t A255806 CoefficientList[Series[E^(3*x/(1-x)), {x, 0, 20}], x] * Range[0, 20]!
%t A255806 Table[If[n==0, 1, 3*(n-1)!*LaguerreL[n-1, 1, -3]], {n, 0, 25}] (* _G. C. Greubel_, Feb 24 2021 *)
%o A255806 (PARI) my(x='x +O('x^50)); Vec(serlaplace(exp(3*x/(1-x)))) \\ _G. C. Greubel_, Feb 05 2017
%o A255806 (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 A255806 (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 A255806 Cf. A000262, A052897, A253286.
%K A255806 nonn,easy,changed
%O A255806 0,2
%A A255806 _Vaclav Kotesovec_, Mar 07 2015