# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a326554
Showing 1-1 of 1

%I A326554 #6 Jul 13 2019 17:04:25
%S A326554 1,2,10,89,1144,19237,402292,10076467,294435680,9842422985,
%T A326554 370678591684,15537544991575,717711797249344,36234873537957421,
%U A326554 1985661659081360852,117415812545786700803,7454037992785099114816,505819653769275584567185,36549387566762559927313924,2802817106895324406986830863,227441704405405503356461103456
%N A326554 E.g.f. A(x) satisfies: A(x) = Sum_{n>=0} (exp(n*x) + A(x))^n * x^n/n!.
%C A326554 More generally, the following sums are equal:
%C A326554 (1) Sum_{n>=0} (q^n + p)^n * r^n / n!,
%C A326554 (2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,
%C A326554 here, q = exp(x), p = A(x), r = x.
%H A326554 Paul D. Hanna, <a href="/A326554/b326554.txt">Table of n, a(n) for n = 0..100</a>
%F A326554 E.g.f. A(x) satisfies:
%F A326554 (1) A(x) = Sum_{n>=0} ( exp(n*x) + A(x) )^n * x^n / n!,
%F A326554 (2) A(x) = Sum_{n>=0} exp(n^2*x) * exp( exp(n*x)*x * A(x) ) * x^n / n!.
%e A326554 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! + ...
%e A326554 such that the following sums are equal
%e A326554 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! + ...
%e A326554 and
%e A326554 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! + ...
%o A326554 (PARI) /* E.g.f. A(x) = Sum_{n>=0} (exp(n*x) + A(x) )^n * x^n/n! */
%o A326554 {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)}
%o A326554 for(n=0,25,print1(a(n),", "))
%o A326554 (PARI) /* E.g.f. A(x) = Sum_{n>=0} exp(n^2*x) * exp( exp(n*x)*x*A(x) )*x^n/n! */
%o A326554 {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)}
%o A326554 for(n=0,25,print1(a(n),", "))
%K A326554 nonn
%O A326554 0,2
%A A326554 _Paul D. Hanna_, Jul 13 2019

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE