A386646 Expansion of e.g.f. Sum_{n>=0} (2^n + x)^n * x^n / n!.

1, 2, 18, 536, 66316, 33636832, 68750980216, 562995064353920, 18446990378410477200, 2417856827427983647531520, 1267651025241922183470966470176, 2658456127743272591813667810372278272, 22300745369876426654206395965130496991176384, 748288839162767087393170357241926671150780067340288

Offset: 0

Author

Paul D. Hanna, Aug 08 2025

Keywords

• nonn

Comments

In general, the following sums are equal:

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

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

here, q = A(x) with p = x, r = x.

Examples

			E.g.f.: A(x) = 1 + 2*x + 18*x^2/2! + 536*x^3/3! + 66316*x^4/4! + 33636832*x^5/5! + 68750980216*x^6/6! + 562995064353920*x^7/7! + 18446990378410477200*x^8/8! + ...
where
A(x) = 1 + (2 + x)*x + (2^2 + x)^2*x^2/2! + (2^3 + x)^3*x^3/n! + (2^4 + x)^4*x^4/4! + (2^5 + x)^5*x^5/5! + ...
Also,
A(x) = exp(x^2) + 2*exp(2*x^2)*x + 2^4*exp(2^2*x^2)*x^2/2! + 2^9*exp(2^3*x^2)*x^3/3! + 2^16*exp(2^4*x^2)*x^4/4! + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..70

Crossrefs

Cf. A136516.

Programs

Mathematica

nmax = 15; CoefficientList[Series[Sum[(2^k + x)^k * x^k / k!, {k, 0, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 09 2025 *)

PARI

{a(n) = my(A = sum(m=0, n, (2^m + x)^m * x^m/m! +x*O(x^n)) ); n!*polcoef(A,n)}
for(n=0, 15, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.

(1) A(x) = Sum_{n>=0} (2^n + x)^n * x^n / n!.

(2) A(x) = Sum_{n>=0} 2^(n^2) * exp(2^n*x^2) * x^n / n!.

a(n) ~ 2^(n^2). - Vaclav Kotesovec, Aug 09 2025