A240165 E.g.f.: exp( x*(1 + exp(2*x)) ).

1, 2, 8, 44, 288, 2192, 18976, 182912, 1934848, 22231808, 275203584, 3645178880, 51370694656, 766634946560, 12066538676224, 199607631945728, 3459736006950912, 62662715180515328, 1183139425871331328, 23237689444403511296, 473852525131782946816, 10014501808427774246912

Offset: 0

Author

Paul D. Hanna, Aug 02 2014

Keywords

• nonn

Examples

			E.g.f.: E(x) = 1 + 2*x + 8*x^2/2! + 44*x^3/3! + 288*x^4/4! + 2192*x^5/5! +...
where E(x) = exp(x) * exp(x*exp(2*x)).
O.g.f.: A(x) = 1 + 2*x + 8*x^2 + 44*x^3 + 288*x^4 + 2192*x^5 +...
where
A(x) = 1/(1-x) + x/(1-3*x)^2 + x^2/(1-5*x)^3 + x^3/(1-7*x)^4 + x^4/(1-9*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.

Programs

Mathematica

Table[Sum[Binomial[n,k] *(2*k+1)^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 06 2014 *)
With[{nn=30},CoefficientList[Series[Exp[x(1+Exp[2x])],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jul 17 2016 *)

PARI

{a(n)=local(A=1);A=exp( x*(1 + exp(2*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 - (2*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, binomial(n,k) * (2*k+1)^(n-k) )}
for(n=0,30,print1(a(n),", "))

Formula

O.g.f.: Sum_{n>=0} x^n / (1 - (2*n+1)*x)^(n+1).

a(n) = Sum_{k=0..n} binomial(n,k) * (2*k+1)^(n-k) for n>=0.

From Vaclav Kotesovec, Aug 06 2014: (Start)

a(n) ~ exp((1+exp(2*r))*r - n) * n^(n+1/2) / (r^n * sqrt(r + exp(2*r)*r*(1 + 6*r + 4*r^2))), where r is the root of the equation r*(1 + exp(2*r) + 2*r*exp(2*r)) = n.

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

(End)