A303291 G.f. A(x) satisfies: 3 = Sum_{n>=0} (2/3)^n * (1 + x)^(n^2) / A(x)^n.

1, 5, 70, 3170, 252160, 27705800, 3806286820, 621124623740, 116766042046000, 24783363325335440, 5854493683431121840, 1522701357625214096240, 432347094526718807347480, 133078785461406479045306360, 44145742694332046133435657280, 15702781293109570148744738306240, 5962874290966165187708554294296880, 2407878412120285331813837276575565360

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

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

G.f.: 3 = Sum_{n>=0} (2/3)^n * (1+x)^n / A(x)^n * Product_{k=1..n} (3*A(x) - 2*(1+x)^(4*k-3)) / (3*A(x) - 2*(1+x)^(4*k-1)), due to a q-series identity.

a(n) ~ 2^(2*n + 2 + log(3/2)/8) * n^n / (3^(log(3/2)/8 + 7/2) * exp(n) * log(3/2)^(2*n + 1)). - Vaclav Kotesovec, Oct 14 2020

Example

G.f.: A(x) = 1 + 5*x + 70*x^2 + 3170*x^3 + 252160*x^4 + 27705800*x^5 + 3806286820*x^6 + 621124623740*x^7 + 116766042046000*x^8 + ...
such that A = A(x) satisfies:
3 = 1 + (1+x)/(3*A/2) + (1+x)^4/(3*A/2)^2 + (1+x)^9/(3*A/2)^3 + (1+x)^16/(3*A/2)^4 + (1+x)^25/(3*A/2)^5 + (1+x)^36/(3*A/2)^6 + (1+x)^49/(3*A/2)^7 + ...

Prog

(PARI) /* Find A(x) that satisfies the continued fraction: */
{a(n) = my(A=[1], q=1+x, CF=1); for(i=1, n, A=concat(A, 0); m=#A; for(k=0, m, CF = 1/(1 - q^(4*m-4*k+1)/(3/2*Ser(A) - q^(2*m-2*k+1)*(q^(2*m-2*k+2) - 1)*CF)) ); A[#A] = Vec(CF)[#A]/6 ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A303290, A303292.

Sequence in context: A174486 A135438 A015502 * A324229 A338989 A101019

Adjacent sequences: A303288 A303289 A303290 * A303292 A303293 A303294

Keyword

nonn

Author

Paul D. Hanna, Apr 22 2018

Status

approved