A245834 E.g.f.: exp( x*(1 + exp(3*x)) ).

1, 2, 10, 71, 592, 5777, 64792, 814025, 11264176, 169871633, 2768582104, 48412950929, 902831609368, 17865749820089, 373564063839376, 8223263706957713, 189960800250512608, 4591950749700004385, 115866075506169417256, 3044877330738661504625, 83169542349597382767496, 2356949307613191494567561

Offset: 0

Author

Paul D. Hanna, Aug 02 2014

Keywords

• nonn

Examples

			E.g.f.: E(x) = 1 + 2*x + 10*x^2/2! + 71*x^3/3! + 592*x^4/4! + 5777*x^5/5! +...
where E(x) = exp(x) * exp(x*exp(3*x)).
O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 71*x^3 + 592*x^4 + 5777*x^5 + 64792*x^6 +...
where
A(x) = 1/(1-x) + x/(1-4*x)^2 + x^2/(1-7*x)^3 + x^3/(1-10*x)^4 + x^4/(1-13*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

Crossrefs

Cf. A080108, A216689, A240165, A245835.

Programs

Mathematica

Table[Sum[Binomial[n,k] *(3*k+1)^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 06 2014 *)
With[{nn=30},CoefficientList[Series[Exp[x(1+Exp[3x])],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jun 09 2019 *)

PARI

{a(n)=local(A=1);A=exp( x*(1 + 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+1)*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+1)^(n-k)*binomial(n,k))}
for(n=0,30,print1(a(n),", "))

Formula

O.g.f.: Sum_{n>=0} x^n / (1 - (3*n+1)*x)^(n+1).

a(n) = Sum_{k=0..n} binomial(n,k) * (3*k+1)^(n-k) for n>=0.

From Vaclav Kotesovec, Aug 06 2014: (Start)

a(n) ~ exp((1+exp(3*r))*r - n) * n^(n+1/2) / (r^n * sqrt(r + exp(3*r)*r* (1+9*r*(1+r)))), where r is the root of the equation r*(1 + exp(3*r) + 3*r*exp(3*r)) = n.

(a(n)/n!)^(1/n) ~ 3*exp(1/(2*LambertW(sqrt(3*n)/2))) / (2*LambertW(sqrt(3*n)/2)).

(End)