A370434 Expansion of Product_{n>=1} (1 - 4^(n-1)*x^n) * (1 + 4^(n-1)*x^n)^2.

1, 1, 3, 19, 60, 348, 1216, 6480, 23040, 121152, 445696, 2214912, 8475648, 40796160, 158564352, 754302976, 2949120000, 13694926848, 55180001280, 250151436288, 1008079994880, 4570684063744, 18552497111040, 82564035379200, 339344829186048, 1494986847682560, 6161930523770880

Offset

0,3

Comments

Compare to Product_{n>=1} (1 - 4^n*x^n) * (1 + 4^n*x^n)^2 = Sum_{n>=0} 4^(n*(n+1)/2) * x^(n*(n+1)/2).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..630

Formula

a(n) ~ c^(1/4) * 2^(2*n + 2) * exp(2*sqrt(c*n)) / (5 * sqrt(3*Pi) * n^(3/4)), where c = -2*polylog(2, -1/4) - polylog(2, 1/4). - Vaclav Kotesovec, Feb 27 2024

Example

G.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 60*x^4 + 348*x^5 + 1216*x^6 + 6480*x^7 + 23040*x^8 + 121152*x^9 + 445696*x^10 + 2214912*x^11 + 8475648*x^12 + ...
where A(x) is the series expansion of the infinite product given by
A(x) = (1 - x)*(1 + x)^2 * (1 - 4*x^2)*(1 + 4*x^2)^2 * (1 - 16*x^3)*(1 + 16*x^3)^2 * (1 - 64*x^4)*(1 + 64*x^4)^2 * ... * (1 - 4^(n-1)*x^n)*(1 + 4^(n-1)*x^n)^2 * ...

Prog

(PARI) {a(n) = polcoeff( prod(k=1, n, (1 - 4^(k-1)*x^k) * (1 + 4^(k-1)*x^k)^2 +x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A370337, A370338, A370435, A344063, A338673.

Sequence in context: A182353 A012863 A269156 * A041015 A378115 A185448

Adjacent sequences: A370431 A370432 A370433 * A370435 A370436 A370437

Keyword

nonn

Author

Paul D. Hanna, Feb 26 2024

Status

approved