A370710 -id:A370710 - cp's OEIS frontend

A370712 a(n) = 3^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/3).

1, 3, 0, 99, -270, 2430, -10287, 105462, -750141, 5702481, -42623901, 347424633, -2779077762, 22353287634, -181730796723, 1493711042589, -12321529794261, 102125312638713, -850797139405887, 7120067746384863, -59800770201017934, 503922807927384129, -4259721779079782751

Offset: 0

Vaclav Kotesovec, Feb 27 2024

sign

Cf. A032308, A370710.

Cf. A300579, A344062.

nmax = 30; CoefficientList[Series[Product[(1 + 3*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 3^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 3*(3*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[(QPochhammer[-3, x]/4)^(1/3), {x, 0, nmax}], x] * 3^Range[0, nmax]

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

a(n) ~ (-1)^(n+1) * c * 9^n / n^(4/3), where c = QPochhammer(-1/3)^(1/3) / (3*Gamma(2/3)) = 0.26286302373105271371291957730496322329245126572...

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

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Feb 28 2024

nonn

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)).

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).

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]

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)).

A370752 a(n) = 3^n * [x^n] Product_{k>=1} ((1 + 3x^k)/(1 - 3x^k))^(1/3).

Original entry on oeis.org

1, 6, 36, 360, 1998, 18792, 121176, 1123632, 7537860, 72078174, 510702408, 4896308088, 35923749480, 345406994280, 2600934294816, 24985346997888, 191735328374478, 1838307293836560, 14317601666954364, 136953233511162840, 1079293961918593800, 10299943344889922832

Offset: 0

Vaclav Kotesovec, Feb 29 2024

nonn

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

Cf. A303390 (d=3,m=1), A370751 (d=3,m=2), A370752 (d=3,m=3).
Cf. A261584 (d=2,m=1), A303346 (d=2,m=2), A370750 (d=2,m=3), A370749 (d=2,m=4).
Cf. A015128 (d=1,m=1), A303307 (d=1,m=2), A303342 (d=1,m=3).
Cf. A032308, A242587.
Cf. A370712, A370710.

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

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

a(n) ~ QPochhammer(-1, 1/3)^(1/3) * 9^n / (Gamma(1/3) * QPochhammer(1/3)^(1/3) * n^(2/3)).

A370734 a(n) = 8^n * [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/4).

Original entry on oeis.org

1, 6, 138, 2292, 47046, 852756, 18266628, 366635112, 7948637382, 170568754692, 3761729402412, 83136335360856, 1863229219846428, 41883396293989320, 948524060727094728, 21555960625992644304, 492036151405623971142, 11264431786398948383844, 258676355450246122857756

Offset: 0

Vaclav Kotesovec, Feb 28 2024

nonn

Cf. A242587 (m=1), A370714 (m=2), A370710 (m=3), A370735 (m=5).

nmax = 20; CoefficientList[Series[Product[1/(1-3*x^k), {k, 1, nmax}]^(1/4), {x, 0, nmax}], x] * 8^Range[0, nmax]
nmax = 20; CoefficientList[Series[Product[1/(1-3*(8*x)^k), {k, 1, nmax}]^(1/4), {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - 3*(8*x)^k)^(1/4).

a(n) ~ 24^n / (Gamma(1/4) * QPochhammer(1/3)^(1/4) * n^(3/4)).

cp's OEIS Frontend

A370712 a(n) = 3^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/3).

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

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Mathematica

Formula

A370752 a(n) = 3^n * [x^n] Product_{k>=1} ((1 + 3x^k)/(1 - 3x^k))^(1/3).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A370734 a(n) = 8^n * [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/4).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A370712 a(n) = 3^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/3).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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

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Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A370752 a(n) = 3^n * [x^n] Product_{k>=1} ((1 + 3*x^k)/(1 - 3*x^k))^(1/3).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A370734 a(n) = 8^n * [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/4).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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

A370752 a(n) = 3^n * [x^n] Product_{k>=1} ((1 + 3x^k)/(1 - 3x^k))^(1/3).