A303058 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (1+x)^(n^2) * x^n / A(x)^n.

1, 1, 1, 2, 5, 16, 61, 259, 1228, 6284, 34564, 201978, 1246652, 8084728, 54862377, 388266809, 2857708840, 21822753453, 172550972216, 1410144139982, 11892084248959, 103343300813517, 924223611649636, 8496346816801059, 80201063980292729, 776585923239589681, 7706568335863727817, 78311132374535936605

Offset: 0

Author

Paul D. Hanna, Apr 20 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 61*x^6 + 259*x^7 + 1228*x^8 + 6284*x^9 + 34564*x^10 + 201978*x^11 + 1246652*x^12 + ...
such that
A(x) = 1 + (1+x)*x/A(x) + (1+x)^4*x^2/A(x)^2 + (1+x)^9*x^3/A(x)^3 + (1+x)^16*x^4/A(x)^4 + (1+x)^25*x^5/A(x)^5 + (1+x)^36*x^6/A(x)^6 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..430

Crossrefs

Cf. A303057, A303290, A301929, A321607, A321608.

Programs

PARI

{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); A[#A] = Vec(sum(n=0,#A, ((1+x)^n +x*O(x^#A))^n * x^n/Ser(A)^n ) )[#A] );A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: A(x) = 1/(1 - q*x/(A(x) - q*(q^2-1)*x/(1 - q^5*x/(A(x) - q^3*(q^4-1)*x/(1 - q^9*x/(A(x) - q^5*(q^6-1)*x/(1 - q^13*x/(A(x) - q^7*(q^8-1)*x/(1 - ...))))))))), where q = (1+x), a continued fraction due to a partial elliptic theta function identity.

G.f.: A(x) = Sum_{n>=0} x^n/A(x)^n * (1+x)^n * Product_{k=1..n} (A(x) - x*(1+x)^(4*k-3)) / (A(x) - x*(1+x)^(4*k-1)), due to a q-series identity.