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A326435 E.g.f.: exp(-4) * Sum_{n>=0} (exp(n*x) + 3)^n / n!. 5
1, 5, 69, 1496, 45771, 1840537, 92925982, 5705543791, 416015394341, 35365673566750, 3454046493504337, 382930667897753421, 47708365129614794580, 6622948820406278058625, 1016977626656613380728781, 171637260767262574245781800, 31661205827344145981298200207, 6352045190999137085697971335893 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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 = 3, r = 1.

LINKS

Table of n, a(n) for n=0..17.

FORMULA

E.g.f.: exp(-4) * Sum_{n>=0} (exp(n*x) + 3)^n / n!.

E.g.f.: exp(-4) * Sum_{n>=0} exp(n^2*x) * exp( 3*exp(n*x) ) / n!.

FORMULAS FOR TERMS.

a(3*n) = 0 (mod 2), a(3*n-1) = 1 (mod 2), and a(3*n-2) = 1 (mod 2) for n > 0.

EXAMPLE

E.g.f.: A(x) = 1 + 5*x + 69*x^2/2! + 1496*x^3/3! + 45771*x^4/4! + 1840537*x^5/5! + 92925982*x^6/6! + 5705543791*x^7/7! + 416015394341*x^8/8! + 35365673566750*x^9/9! + 3454046493504337*x^10/10! + ...

such that

A(x) = exp(-4) * (1 + (exp(x) + 3) + (exp(2*x) + 3)^2/2! + (exp(3*x) + 3)^3/3! + (exp(4*x) + 3)^4/4! + (exp(5*x) + 3)^5/5! + (exp(6*x) + 3)^6/6! + ...)

also

A(x) = exp(-4) * (exp(3) + exp(x)*exp(3*exp(x)) + exp(4*x)*exp(3*exp(2*x))/2! + exp(9*x)*exp(3*exp(3*x))/3! + exp(16*x)*exp(3*exp(4*x))/4! + exp(25*x)*exp(3*exp(5*x))/5! + exp(36*x)*exp(3*exp(6*x))/6! + ...).

PROG

(PARI) /* Requires suitable precision */

\p200

Vec(round(serlaplace( exp(-4) * sum(n=0, 500, (exp(n*x +O(x^31)) + 3)^n/n! ) )))

CROSSREFS

Cf. A326600, A020557, A326433, A326434, A326436, A326437.

Sequence in context: A333035 A333459 A307700 * A231294 A302543 A157038

Adjacent sequences:  A326432 A326433 A326434 * A326436 A326437 A326438

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 11 2019

STATUS

approved

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Last modified August 7 23:50 EDT 2022. Contains 355995 sequences. (Running on oeis4.)