A301927 G.f. A(x) satisfies: x = Sum_{n>=1} x^n / ( (1-x)^(n^2) * A(x)^n ).

1, 2, 4, 9, 24, 77, 294, 1296, 6403, 34644, 201932, 1253513, 8219110, 56578239, 406990651, 3048202700, 23700070773, 190830842843, 1588016365186, 13633603416558, 120574656241999, 1097006289005674, 10255338612462641, 98403208150304070, 968186766428157206, 9759036265967791137, 100690787844977985900, 1062601625749170026894, 11461320511629994319890

Offset: 0

Author

Paul D. Hanna, May 06 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 9*x^3 + 24*x^4 + 77*x^5 + 294*x^6 + 1296*x^7 + 6403*x^8 + 34644*x^9 + 201932*x^10 + 1253513*x^11 + 8219110*x^12 + ...
such that
x = x/((1-x)*A(x)) + x^2/((1-x)^4*A(x)^2) + x^3/((1-x)^9*A(x)^3) + x^4/((1-x)^16*A(x)^4) + x^5/((1-x)^25*A(x)^5) + x^6/((1-x)^36*A(x)^6) + x^7/((1-x)^49*A(x)^7) + x^8/((1-x)^64*A(x)^8) + ...
		

Crossrefs

Cf. A301929.

Programs

PARI

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

Formula

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

G.f.: 1+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/(1-x), a continued fraction due to a partial elliptic theta function identity.