A370435 Expansion of Product_{n>=1} (1 - 5^(n-1)*x^n) * (1 + 5^(n-1)*x^n)^2.

1, 1, 4, 29, 120, 820, 3625, 23400, 105000, 669500, 3075625, 18837500, 89237500, 532500000, 2554062500, 15086640625, 72843750000, 421773437500, 2084812500000, 11834804687500, 58638281250000, 332210205078125, 1656773437500000, 9240966796875000, 46624682617187500, 257479980468750000

Offset: 0

Author

Paul D. Hanna, Feb 26 2024

Keywords

• nonn

Comments

Compare to Product_{n>=1} (1 - 5^n*x^n) * (1 + 5^n*x^n)^2 = Sum_{n>=0} 5^(n*(n+1)/2) * x^(n*(n+1)/2).

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 120*x^4 + 820*x^5 + 3625*x^6 + 23400*x^7 + 105000*x^8 + 669500*x^9 + 3075625*x^10 + 18837500*x^11 + ...
where A(x) is the series expansion of the infinite product given by
A(x) = (1 - x)*(1 + x)^2 * (1 - 5*x^2)*(1 + 5*x^2)^2 * (1 - 25*x^3)*(1 + 25*x^3)^2 * (1 - 125*x^4)*(1 + 125*x^4)^2 * ... * (1 - 5^(n-1)*x^n)*(1 + 5^(n-1)*x^n)^2 * ...
		

Links

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

Crossrefs

Cf. A370337, A370338, A370434, A344064, A338674.

Programs

PARI

{a(n) = polcoeff( prod(k=1,n, (1 - 5^(k-1)*x^k) * (1 + 5^(k-1)*x^k)^2 +x*O(x^n)), n)}
for(n=0,40, print1(a(n),", "))

Formula

a(n) ~ c^(1/4) * 5^(n + 3/2) * exp(2*sqrt(c*n)) / (24 * sqrt(Pi) * n^(3/4)), where c = -2*polylog(2, -1/5) - polylog(2, 1/5). - Vaclav Kotesovec, Feb 27 2024