A380687 G.f. satisfies A(x) = Sum_{n>=0} x^n * (1+x)^(2*n^2) / A(x)^(n*(n+1)).

1, 1, 1, 1, 2, 3, 16, 50, 270, 1266, 7017, 40073, 243665, 1556727, 10394483, 72403198, 524195255, 3936080686, 30591479055, 245655102572, 2035083634357, 17369029459700, 152535273188651, 1376833447490442, 12760355850982450, 121311719667445606, 1182004361469302527, 11793836041463723717, 120413981027066126060

Offset

0,5

Comments

Compare to F(x) = Sum_{n>=0} x^n * (1+x)^(n^2) / F(x)^(n*(n+1)) holds when F(x) = (1+x).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 16*x^6 + 50*x^7 + 270*x^8 + 1266*x^9 + 7017*x^10 + 40073*x^11 + 243665*x^12 + ...

where

A(x) = 1 + x*(1+x)^2/A(x)^2 + x^2*(1+x)^8/A(x)^6 + x^3*(1+x)^18/A(x)^12 + x^4*(1+x)^32/A(x)^20 + x^5*(1+x)^50/A(x)^30 + x^6*(1+x)^72/A(x)^42 + x^7*(1+x)^98/A(x)^56 + x^8*(1+x)^128/A(x)^72 + x^9*(1+x)^162/A(x)^90 + ...

Prog

(PARI) {a(n) = my(V=[1]); for(i=1, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=0, #A, x^m*(1+x +x*O(x^#A))^(2*m^2) / A^(m*(m+1))) - A, #V-1) ); H=A; V[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A321099, A380679.

Sequence in context: A012352 A012704 A012698 * A302611 A012355 A012353

Adjacent sequences: A380684 A380685 A380686 * A380690 A380691 A380692

Keyword

nonn

Author

Paul D. Hanna, Feb 23 2025

Status

approved