A325060 - OEIS

%I #7 Apr 12 2019 17:15:16

%S 1,1,2,15,148,2565,59046,1825831,70678280,3343670217,188213143690,

%T 12380664239691,937445644041996,80731184378264173,7828455595505947598,

%U 847603141493494555695,101732530008690207859216,13451340197177805355768209,1948644186311260903900163346,307791516722206533702105826963,52778747788778673408416550382100

%N E.g.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * exp(n^2*x) / A(x)^n.

%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 15*x^3/3! + 148*x^4/4! + 2565*x^5/5! + 59046*x^6/6! + 1825831*x^7/7! + 70678280*x^8/8! + 3343670217*x^9/9! + 188213143690*x^10/10! + ...

%e such that

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

%e Note that a(n) is divisible by n, for n >= 1, where a(n)/n starts

%e [1, 1, 5, 37, 513, 9841, 260833, 8834785, 371518913, 18821314369, ...].

%e RELATED SERIES.

%e log(A(x)) =  x + x^2/2! + 11*x^3/3! + 94*x^4/4! + 1849*x^5/5! + 42966*x^6/6! + 1385509*x^7/7! + 54885832*x^8/8! + 2654774721*x^9/9! + 152054810650*x^10/10! + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);

%o A[#A] = polcoeff( sum(m=0,#A, x^m * exp(m^2*x +x*O(x^n)) / Ser(A)^(m+1)), #A-1)); n!*A[n+1]}

%o for(n=0,30, print1(a(n),", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 12 2019