A370739 a(n) = 5^(2*n) * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/5).

1, 15, -75, 35250, -1138125, 72645000, -3307996875, 244578890625, -15502648125000, 985908809765625, -63515254624218750, 4314500023927734375, -291905297026816406250, 19789483493484814453125, -1355414138248614990234375, 93666904586649390380859375, -6498800175020013123779296875

Offset: 0

Author

Vaclav Kotesovec, Feb 28 2024

Keywords

• sign

Comments

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

Crossrefs

Cf. A032308 (d=3,m=1), A370711 (d=3,m=2), A370712 (d=3,m=3), A370738 (d=3,m=4).

Cf. A032302 (d=2,m=1), A370709 (d=2,m=2), A370716 (d=2,m=3), A370736 (d=2,m=4), A370737 (d=2,m=5).

Cf. A000009 (d=1,m=1), A298994 (d=1,m=2), A303074 (d=1,m=3)

Programs

Mathematica

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

Formula

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

a(n) ~ (-1)^(n+1) * QPochhammer(-1/3)^(1/5) * 75^n / (5 * Gamma(4/5) * n^(6/5)).