A370735 a(n) = 5^(2*n) * [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/5).

1, 15, 1050, 52125, 3277500, 179801250, 11966690625, 738318187500, 49788716718750, 3314446448437500, 227432073022265625, 15631633385109375000, 1090877899335878906250, 76338563689129101562500, 5384934139819611328125000, 381204340327212964599609375, 27111589537137988341064453125

Offset: 0

Author

Vaclav Kotesovec, Feb 28 2024

Keywords

• nonn

Comments

In general, if d > 1, m >= 1 and g.f. = Product_{k>=1} 1/(1 - d*x^k)^(1/m), then a(n) ~ d^n / (Gamma(1/m) * QPochhammer(1/d)^(1/m) * n^(1 - 1/m)).

Crossrefs

Cf. A242587 (d=3,m=1), A370714 (d=3,m=2), A370710 (d=3,m=3), A370734 (d=3,m=4).

Cf. A070933 (d=2,m=1), A370713 (d=2,m=2), A370715 (d=2,m=3), A370732 (d=2,m=4), A370733 (d=2,m=5).

Cf. A000041 (d=1,m=1), A271235 (d=1,m=2), A271236 (d=1,m=3).

Programs

Mathematica

nmax = 20; CoefficientList[Series[Product[1/(1-3*x^k), {k, 1, nmax}]^(1/5), {x, 0, nmax}], x] * 25^Range[0, nmax]
nmax = 20; CoefficientList[Series[Product[1/(1-3*(25*x)^k), {k, 1, nmax}]^(1/5), {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - 3*(25*x)^k)^(1/5).

a(n) ~ 75^n / (Gamma(1/5) * QPochhammer(1/3)^(1/5) * n^(4/5)).