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E.g.f. A(x) satisfies: A(x) = Sum_{n>=0} (exp(n*x) + A(x))^n * x^n/n!.
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%I #6 Jul 13 2019 17:04:25

%S 1,2,10,89,1144,19237,402292,10076467,294435680,9842422985,

%T 370678591684,15537544991575,717711797249344,36234873537957421,

%U 1985661659081360852,117415812545786700803,7454037992785099114816,505819653769275584567185,36549387566762559927313924,2802817106895324406986830863,227441704405405503356461103456

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

%C More generally, the following sums are equal:

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

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

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

%H Paul D. Hanna, <a href="/A326554/b326554.txt">Table of n, a(n) for n = 0..100</a>

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

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

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

%e 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 such that the following sums are equal

%e 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 and

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

%o {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 for(n=0,25,print1(a(n),", "))

%o (PARI) /* E.g.f. A(x) = Sum_{n>=0} exp(n^2*x) * exp( exp(n*x)*x*A(x) )*x^n/n! */

%o {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 for(n=0,25,print1(a(n),", "))

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 13 2019