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

1, 3, 15, 99, 801, 7623, 83079, 1017495, 13808097, 205374123, 3318673599, 57845821707, 1081091446785, 21553820597871, 456410531639799, 10225931132021247, 241609515712343361, 6002109578246918355, 156360266121378584943, 4261404847790207796147

Offset: 0

Author

Vaclav Kotesovec, Mar 07 2015

Keywords

• nonn
• easy
• changed

Comments

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)).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..435
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A000262, A052897, A253286.

Programs

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

Mathematica

nmax=20; CoefficientList[Series[Exp[Sum[3*x^k,{k,1,nmax}]],{x,0,nmax}],x] * Range[0,nmax]!
CoefficientList[Series[E^(3*x/(1-x)), {x, 0, 20}], x] * Range[0, 20]!
Table[If[n==0, 1, 3*(n-1)!*LaguerreL[n-1, 1, -3]], {n, 0, 25}] (* G. C. Greubel, Feb 24 2021 *)

PARI

my(x='x +O('x^50)); Vec(serlaplace(exp(3*x/(1-x)))) \\ G. C. Greubel, Feb 05 2017

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

Formula

E.g.f.: exp(3*x/(1-x)).

a(n) ~ 3^(1/4) * exp(2*sqrt(3*n) - 3/2) * n! / (2*sqrt(Pi)*n^(3/4)).

a(n) = (2*n+1)*a(n-1) - (n-2)*(n-1)*a(n-2). - Vaclav Kotesovec, Nov 04 2016

From G. C. Greubel, Feb 24 2021: (Start)

a(n) = A253286(n+3, 3).

a(n) = 3*(n-1)!*LaguerreL(n-1, 1, -3) with a(0) = 1. (End)

For n > 0, a(n) = (n-1)! * Sum_{k=1..n} binomial(n,k) * 3^k / (k-1)!. - Vaclav Kotesovec, Aug 24 2025