A245835 E.g.f.: exp( x*(2 + exp(3*x)) ).
1, 3, 15, 108, 945, 9558, 109917, 1412316, 19959777, 306805482, 5087064789, 90370321704, 1710170426097, 34308056537550, 726612812416269, 16188742781216892, 378244417385086785, 9242436410233527762, 235609985190361119525, 6252379688953421699760, 172380307421633200750161
Offset: 0
Keywords
Examples
E.g.f.: E(x) = 1 + 3*x + 15*x^2/2! + 108*x^3/3! + 945*x^4/4! + 9558*x^5/5! +... where E(x) = exp(2*x) * exp(x*exp(3*x)). O.g.f.: A(x) = 1 + 3*x + 15*x^2 + 108*x^3 + 945*x^4 + 9558*x^5 + 109917*x^6 +... where A(x) = 1/(1-2*x) + x/(1-5*x)^2 + x^2/(1-8*x)^3 + x^3/(1-11*x)^4 + x^4/(1-14*x)^5 +...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function
Programs
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Mathematica
Table[Sum[Binomial[n,k] * (3*k+2)^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 03 2014 *) With[{nn=20},CoefficientList[Series[Exp[x(2+Exp[3x])],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 06 2015 *) -
PARI
{a(n)=local(A=1);A=exp( x*(2 + exp(3*x +x*O(x^n))) );n!*polcoeff(A, n)} for(n=0,30,print1(a(n),", ")) -
PARI
{a(n)=local(A=1);A=sum(k=0, n, x^k/(1 - (3*k+2)*x +x*O(x^n))^(k+1));polcoeff(A, n)} for(n=0,30,print1(a(n),", ")) -
PARI
{a(n)=sum(k=0,n,(3*k+2)^(n-k)*binomial(n,k))} for(n=0,30,print1(a(n),", "))
Formula
O.g.f.: Sum_{n>=0} x^n / (1 - (3*n+2)*x)^(n+1).
a(n) = Sum_{k=0..n} binomial(n,k) * (3*k+2)^(n-k) for n>=0.
a(n) ~ exp((n+6*r^2)/(1+3*r)) * n! / (r^n*sqrt(2*Pi*(-6*r^2*(2+3*r) + n*(1+9*r+9*r^2)) / (1+3*r))), where r is the root of the equation r*(2 + (1+3*r)*exp(3*r)) = n. - Vaclav Kotesovec, Aug 03 2014
(a(n)/n!)^(1/n) ~ 3*exp(1/(2*LambertW(sqrt(3*n)/2))) / (2*LambertW(sqrt(3*n)/2)). - Vaclav Kotesovec, Aug 06 2014