A326430 E.g.f.: exp(-1) * Sum_{n>=0} (exp(n*x) + x)^n / n!.

1, 3, 22, 297, 6055, 169431, 6145827, 277912452, 15225719420, 988814989679, 74822364609113, 6505084496930641, 642317112612827029, 71331999557857791694, 8835651007377368848464, 1211946040741011512724559, 182930472229597183037431011, 30216143201862939999461382959, 5435054718681965118312689633935

Offset: 0

Author

Paul D. Hanna, Jul 09 2019

Keywords

• nonn

Comments

More generally, the following sums are equal:

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

(2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,

here, q = exp(x), p = x, r = 1.

Examples

			E.g.f.: A(x) = 1 + 3*x + 22*x^2/2! + 297*x^3/3! + 6055*x^4/4! + 169431*x^5/5! + 6145827*x^6/6! + 277912452*x^7/7! + 15225719420*x^8/8! + 988814989679*x^9/9! + 74822364609113*x^10/10! + ...
such that
A(x) = exp(-1) * (1 + (exp(x) + x) + (exp(2*x) + x)^2/2! + (exp(3*x) + x)^3/3! + (exp(4*x) + x)^4/4! + (exp(5*x) + x)^5/5! + (exp(6*x) + x)^6/6! + (exp(7*x) + x)^7/7! + (exp(8*x) + x)^8/8! + ...)
also,
A(x) = exp(-1) * (exp(x) + exp(x)*exp(x*exp(x)) + exp(4*x)*exp(x*exp(2*x))/2! + exp(9*x)*exp(x*exp(3*x))/3! + exp(16*x)*exp(x*exp(4*x))/4! + exp(25*x)*exp(x*exp(5*x))/5! + exp(36*x)*exp(x*exp(6*x))/6! + ...).
		

Links

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

Crossrefs

Cf. A326433.

Programs

PARI

/* Requires appropriate precision */
\p200
{a(n) = my(A = exp(-1) * sum(m=0,n+300, (exp(m*x +x*O(x^n)) + x)^m / m! )); round(n!*polcoeff(A,n))}
for(n=0,20,print1(a(n),", "))

Formula

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

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