A303292 G.f. A(x) satisfies: 4 = Sum_{n>=0} (3/4)^n * (1 + x)^(n^2) / A(x)^n.

1, 7, 189, 17283, 2755053, 604260531, 165416203197, 53736069429315, 20098682471065149, 8484270818691168963, 3985069388942026022589, 2060504358592580623699011, 1162904612283296975554475517, 711422819982429170172765550083, 469007739834268780510389856367613, 331521891387779056571085490125831171, 250157485456407234540581483486760865533

Offset

0,2

Links

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

Formula

G.f.: 4 = 1/(1 - q/(4/3*A(x) - q*(q^2-1)/(1 - q^5/(4/3*A(x) - q^3*(q^4-1)/(1 - q^9/(4/3*A(x) - q^5*(q^6-1)/(1 - q^13/(4/3*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.: 4 = Sum_{n>=0} (3/4)^n * (1+x)^n / A(x)^n * Product_{k=1..n} (4*A(x) - 3*(1+x)^(4*k-3)) / (4*A(x) - 3*(1+x)^(4*k-1)), due to a q-series identity.

a(n) ~ 3^(5/2) * 2^(2*n - 17/2) * n^n / (exp(n + log(4/3)^2 / 8) * log(4/3)^(2*n + 1)). - Vaclav Kotesovec, Oct 14 2020

Example

G.f.: A(x) = 1 + 7*x + 189*x^2 + 17283*x^3 + 2755053*x^4 + 604260531*x^5 + 165416203197*x^6 + 53736069429315*x^7 + 20098682471065149*x^8 + ...
such that A = A(x) satisfies:
4 = 1 + (1+x)/(4*A/3) + (1+x)^4/(4*A/3)^2 + (1+x)^9/(4*A/3)^3 + (1+x)^16/(4*A/3)^4 + (1+x)^25/(4*A/3)^5 + (1+x)^36/(4*A/3)^6 + (1+x)^49/(4*A/3)^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)/(4/3*Ser(A) - q^(2*m-2*k+1)*(q^(2*m-2*k+2) - 1)*CF)) ); A[#A] = Vec(CF)[#A]/12 ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A303290, A303291.

Sequence in context: A219567 A202791 A304859 * A010332 A368592 A198258

Adjacent sequences: A303289 A303290 A303291 * A303293 A303294 A303295

Keyword

nonn

Author

Paul D. Hanna, Apr 22 2018

Status

approved