A326432 E.g.f.: exp(-3) * Sum_{n>=0} ((1+x)^n + 2)^n / n!.

1, 4, 43, 762, 19573, 672374, 29390733, 1578973510, 101612589283, 7679375658354, 670906936259299, 66891320576455142, 7530075312966689409, 948460025747139087802, 132635012110499511683869, 20454728573277460691412006, 3458323793329321035116835859, 637694404371402843143395980434, 127650318560095585201739965521651

Offset

0,2

Comments

More generally, the following sums are equal:

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

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

here, q = 1+x, p = 2, r = 1.

Links

Table of n, a(n) for n=0..18.

Formula

E.g.f.: exp(-3) * Sum_{n>=0} ((1+x)^n + 2)^n / n!.

E.g.f.: exp(-3) * Sum_{n>=0} (1+x)^(n^2) * exp( 2*(1+x)^n ) / n!.

Example

E.g.f.: A(x) = 1 + 4*x + 43*x^2/2! + 762*x^3/3! + 19573*x^4/4! + 672374*x^5/5! + 29390733*x^6/6! + 1578973510*x^7/7! + 101612589283*x^8/8! + 7679375658354*x^9/9! + 670906936259299*x^10/10! + ...
such that
A(x) = exp(-3) * (1 + ((1+x) + 2) + ((1+x)^2 + 2)^2/2! + ((1+x)^3 + 2)^3/3! + ((1+x)^4 + 2)^4/4! + ((1+x)^5 + 2)^5/5! + ((1+x)^6 + 2)^6/6! + ...)
also,
A(x) = exp(-3) * (exp(2) + (1+x)*exp(2*(1+x)) + (1+x)^4*exp(2*(1+x)^2)/2! + (1+x)^9*exp(2*(1+x)^3)/3! + (1+x)^16*exp(2*(1+x)^4)/4! + (1+x)^25*exp(2*(1+x)^5)/5! + (1+x)^36*exp(2*(1+x)^6)/6! + ...).

Prog

(PARI) /* Requires appropriate precision */
\p200
{a(n) = my(A = exp(-3) * sum(m=0, n+300, ((1+x)^m + 2 +x*O(x^n))^m / m! )); round(n!*polcoeff(A, n))}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A326431.

Sequence in context: A027311 A198205 A277639 * A074702 A197717 A277456

Adjacent sequences: A326429 A326430 A326431 * A326433 A326434 A326435

Keyword

nonn

Author

Paul D. Hanna, Jul 09 2019

Status

approved