A326091 E.g.f.: Sum_{n>=0} (2 + exp(n*x))^n * x^n/n!.

1, 3, 11, 66, 601, 7418, 116505, 2248522, 52025473, 1414524690, 44471074249, 1595792690594, 64659403375137, 2931455146804330, 147550017664392457, 8189594420467104042, 498288959815836863233, 33061714451161940667554, 2381086262720126177230473, 185362512554618232339122578, 15539467373234774634135507361, 1398111233425766921500901239098, 134584560980879138160145116701257

Offset: 0

Author

Paul D. Hanna, Jun 28 2019

Keywords

• nonn

Comments

More generally, the following sums are equal:

(1) Sum_{n>=0} (p + q^n)^n * r^n/n!,

(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;

here, q = exp(x) with p = 2, r = x.

Examples

			E.g.f.: A(x) = 1 + 3*x + 11*x^2/2! + 66*x^3/3! + 601*x^4/4! + 7418*x^5/5! + 116505*x^6/6! + 2248522*x^7/7! + 52025473*x^8/8! + 1414524690*x^9/9! + 44471074249*x^10/10! + ...
such that
A(x) = 1 + (2 + exp(x))*x + (2 + exp(2*x))^2*x^2/2! + (2 + exp(3*x))^3*x^3/3! + (2 + exp(4*x))^4*x^4/4! + (2 + exp(5*x))^5*x^5/5! + (2 + exp(6*x))^6*x^6/6! + ...
also
A(x) = exp(2*x) + exp(x + 2*exp(x)*x)*x + exp(4*x + 2*exp(2*x)*x)*x^2/2! + exp(9*x + 2*exp(3*x)*x)*x^3/3! + exp(16*x + 2*exp(4*x)*x)*x^4/4! + exp(25*x + 2*exp(5*x)*x)*x^5/5! + exp(36*x + 2*exp(6*x)*x)*x^6/6! + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A108459, A326090, A326261.

Programs

PARI

/* E.g.f.: Sum_{n>=0} (2 + exp(n*x))^n * x^n/n! */
{a(n) = my(A = sum(m=0, n, (2 + exp(m*x +x*O(x^n)))^m * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

PARI

/* E.g.f.: Sum_{n>=0} exp( n^2*x + 2*exp(n*x)*x ) * x^n/n! */
{a(n) = my(A = sum(m=0, n, exp(m^2*x + 2*exp(m*x +x*O(x^n))*x ) * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f.: Sum_{n>=0} (2 + exp(n*x))^n * x^n/n!.

E.g.f.: Sum_{n>=0} exp(n^2*x) * exp( 2*exp(n*x)*x ) * x^n/n!.