A326554 E.g.f. A(x) satisfies: A(x) = Sum_{n>=0} (exp(n*x) + A(x))^n * x^n/n!.

1, 2, 10, 89, 1144, 19237, 402292, 10076467, 294435680, 9842422985, 370678591684, 15537544991575, 717711797249344, 36234873537957421, 1985661659081360852, 117415812545786700803, 7454037992785099114816, 505819653769275584567185, 36549387566762559927313924, 2802817106895324406986830863, 227441704405405503356461103456

Offset

0,2

Comments

More generally, the following sums are equal:

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

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

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

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Formula

E.g.f. A(x) satisfies:

(1) A(x) = Sum_{n>=0} ( exp(n*x) + A(x) )^n * x^n / n!,

(2) A(x) = Sum_{n>=0} exp(n^2*x) * exp( exp(n*x)*x * A(x) ) * x^n / n!.

Example

E.g.f.: A(x) = 1 + 2*x + 10*x^2/2! + 89*x^3/3! + 1144*x^4/4! + 19237*x^5/5! + 402292*x^6/6! + 10076467*x^7/7! + 294435680*x^8/8! + 9842422985*x^9/9! + 370678591684*x^10/10! + ...
such that the following sums are equal
A(x) = 1 + (exp(x) + A(x)) + (exp(2*x) + A(x))^2*x^2/2! + (exp(3*x) + A(x))^3*x^3/3! + (exp(4*x) + A(x))^4*x^4/4! + (exp(5*x) + A(x))^5*x^5/5! + ...
and
A(x) = exp(x*A(x)) + exp(x)*exp(exp(x)*x*A(x))*x + exp(4*x)*exp(exp(2*x)*x*A(x))*x^2/2! + exp(9*x)*exp(exp(3*x)*x*A(x))*x^3/3! + exp(16*x)*exp(exp(4*x)*x*A(x))*x^4/4! + ...

Prog

(PARI) /* E.g.f. A(x) = Sum_{n>=0} (exp(n*x) + A(x) )^n * x^n/n! */
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, (exp(m*x +x*O(x^n)) + A)^m*x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* E.g.f. A(x) = Sum_{n>=0} exp(n^2*x) * exp( exp(n*x)*x*A(x) )*x^n/n! */
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, #A, exp(m^2*x + exp(m*x +x*O(x^n))*x * A)*x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Sequence in context: A096658 A346371 A186184 * A055779 A363425 A338050

Adjacent sequences: A326551 A326552 A326553 * A326555 A326556 A326557

Keyword

nonn

Author

Paul D. Hanna, Jul 13 2019

Status

approved