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A303654 G.f. A(x) satisfies: x = Sum_{n>=1} ((1+x)^n - 1)^n / A(x)^n.

%I A303654 #15 Jun 09 2025 04:23:42
%S A303654 1,4,15,110,1319,21626,440406,10567338,289567071,8881182506,
%T A303654 300677809182,11123151634732,446124758009229,19278179994562787,
%U A303654 892894885467043254,44130236756271133940,2318466084626196805383,129037338117765390406606,7585054768393048247917776,469612308887467564648834414
%N A303654 G.f. A(x) satisfies: x = Sum_{n>=1} ((1+x)^n - 1)^n / A(x)^n.
%H A303654 Paul D. Hanna, <a href="/A303654/b303654.txt">Table of n, a(n) for n = 0..100</a>
%F A303654 G.f.: A(x) = (1+x) / ( Sum_{n>=0} (1+x)^(n^2) / ((1+x)^n + A(x))^(n+1) ).
%F A303654 a(n) ~ c * d^n * n! * sqrt(n), where d = A317855 = 3.161088... and c = 0.292671... - _Vaclav Kotesovec_, Jun 09 2025
%e A303654 G.f.: A(x) = 1 + 4*x + 15*x^2 + 110*x^3 + 1319*x^4 + 21626*x^5 + 440406*x^6 + 10567338*x^7 + 289567071*x^8 + 8881182506*x^9 + ...
%e A303654 such that
%e A303654 x = ((1+x) - 1)/A(x) + ((1+x)^2 - 1)^2/A(x)^2 + ((1+x)^3 - 1)^3/A(x)^3 + ((1+x)^4 - 1)^4/A(x)^4 + ((1+x)^5 - 1)^5/A(x)^5 + ((1+x)^6 - 1)^6/A(x)^6 + ...
%e A303654 Also,
%e A303654 A(x) = (1+x) / ( 1/(1 + A(x)) + (1+x)/((1+x) + A(x))^2 + (1+x)^4/((1+x)^2 + A(x))^3 + (1+x)^9/((1+x)^3 + A(x))^4 + (1+x)^16/((1+x)^4 + A(x))^5 + (1+x)^25/((1+x)^5 + A(x))^6 + (1+x)^36/((1+x)^6 + A(x))^7 + ... ).
%o A303654 (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec(sum(m=1, #A, ((1+x)^m - 1 +x*O(x^#A))^m / Ser(A)^m ) )[#A] ); A[n+1]}
%o A303654 for(n=0, 30, print1(a(n), ", "))
%K A303654 nonn
%O A303654 0,2
%A A303654 _Paul D. Hanna_, May 04 2018