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

1, 4, 15, 110, 1319, 21626, 440406, 10567338, 289567071, 8881182506, 300677809182, 11123151634732, 446124758009229, 19278179994562787, 892894885467043254, 44130236756271133940, 2318466084626196805383, 129037338117765390406606, 7585054768393048247917776, 469612308887467564648834414

Offset: 0

Author

Paul D. Hanna, May 04 2018

Keywords

• nonn

Examples

			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 + ...
such that
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 + ...
Also,
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 + ... ).
		

Links

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

Programs

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]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: A(x) = (1+x) / ( Sum_{n>=0} (1+x)^(n^2) / ((1+x)^n + A(x))^(n+1) ).

a(n) ~ c * d^n * n! * sqrt(n), where d = A317855 = 3.161088... and c = 0.292671... - Vaclav Kotesovec, Jun 09 2025