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A326261 E.g.f.: Sum_{n>=0} (3 + exp(n*x))^n * x^n/n!. 3
1, 4, 18, 115, 1076, 13749, 223342, 4437115, 105308472, 2930229721, 94110395546, 3444510650343, 142161931150564, 6557368148307253, 335460464343013494, 18907437932151629899, 1167279375125285092592, 78529603970775837111729, 5730854443905658384812466, 451803953552256670477653679, 38337003901469883140928003036, 3489532046271886600931347767373 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

More generally, the following sums are equal:

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

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

here, q = exp(x) with p = 3, r = x.

LINKS

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

FORMULA

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

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

EXAMPLE

E.g.f.: A(x) = 1 + 4*x + 18*x^2/2! + 115*x^3/3! + 1076*x^4/4! + 13749*x^5/5! + 223342*x^6/6! + 4437115*x^7/7! + 105308472*x^8/8! + 2930229721*x^9/9! + 94110395546*x^10/10! + ...

such that

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

also

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

PROG

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

{a(n) = my(A = sum(m=0, n, (3 + exp(m*x +x*O(x^n)))^m * x^m/m! )); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

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

{a(n) = my(A = sum(m=0, n, exp(m^2*x + 3*exp(m*x +x*O(x^n))*x ) * x^m/m! )); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A108459, A326090, A326091.

Sequence in context: A141714 A220223 A137567 * A277473 A278994 A223008

Adjacent sequences:  A326258 A326259 A326260 * A326262 A326263 A326264

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 29 2019

STATUS

approved

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Last modified September 25 20:01 EDT 2021. Contains 347659 sequences. (Running on oeis4.)