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A326432 E.g.f.: exp(-3) * Sum_{n>=0} ((1+x)^n + 2)^n / n!. 1
1, 4, 43, 762, 19573, 672374, 29390733, 1578973510, 101612589283, 7679375658354, 670906936259299, 66891320576455142, 7530075312966689409, 948460025747139087802, 132635012110499511683869, 20454728573277460691412006, 3458323793329321035116835859, 637694404371402843143395980434, 127650318560095585201739965521651 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 1 04:31 EDT 2022. Contains 357134 sequences. (Running on oeis4.)