A300581 -id:A300581 - cp's OEIS frontend

A370709 a(n) = 2^n * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/2).

1, 2, 2, 20, 6, 108, 148, 776, -186, 5964, -4, 51032, -89700, 512120, -1259416, 6406032, -19733434, 78363148, -268823572, 1047941688, -3800035916, 14327505832, -52766730600, 199492430192, -746479735524, 2811936761016, -10588174502568, 40092283176560, -151796846803592

Offset: 0

Vaclav Kotesovec, Feb 27 2024

sign

Cf. A032302, A370713.

Cf. A075900, A298994, A300581, A304961, A327550.

nmax = 30; CoefficientList[Series[Product[(1 + 2*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 2^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 2*(2*x)^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Sqrt[QPochhammer[-2, x]/3], {x, 0, nmax}], x] * 2^Range[0, nmax]

G.f.: Product_{k>=1} (1 + 2*(2*x)^k)^(1/2).

a(n) ~ (-1)^(n+1) * c * 4^n / n^(3/2), where c = QPochhammer(-1/2)^(1/2) / (2*sqrt(Pi)) = 0.31039710860287467176143051675437...

A370716 a(n) = 3^(2n) [x^n] Product_{k>=1} (1 + 2*x^k)^(1/3).

Original entry on oeis.org

1, 6, 18, 1170, -1890, 133326, 101250, 20498994, -164656314, 3778220862, -28085954094, 771567716970, -10691904063114, 183594050113518, -2711145260068326, 49416883617381354, -789899109743435994, 13176840267952166070, -216403389726994588086, 3681309971143060236810

Offset: 0

Vaclav Kotesovec, Feb 27 2024

sign

Cf. A032302, A370715.

Cf. A075900, A300581, A304961, A327550.

nmax = 20; CoefficientList[Series[Product[(1 + 2*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 3^(2*Range[0, nmax])
nmax = 20; CoefficientList[Series[Product[(1 + 2*(9*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]
nmax = 20; CoefficientList[Series[(QPochhammer[-2, x]/3)^(1/3), {x, 0, nmax}], x] * 3^(2*Range[0, nmax])

G.f.: Product_{k>=1} (1 + 2*(9*x)^k)^(1/3).

a(n) ~ (-1)^(n+1) * c * 18^n / n^(4/3), where c = QPochhammer(-1/2)^(1/3) / (3*Gamma(2/3)) = 0.2623638446186535909018671540030519...

cp's OEIS Frontend

A370709 a(n) = 2^n * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/2).

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Author

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Mathematica

Formula

A370716 a(n) = 3^(2n) [x^n] Product_{k>=1} (1 + 2*x^k)^(1/3).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A370709 a(n) = 2^n * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A370716 a(n) = 3^(2*n) * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/3).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A370716 a(n) = 3^(2n) [x^n] Product_{k>=1} (1 + 2*x^k)^(1/3).