cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A370711 a(n) = 4^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/2).

Original entry on oeis.org

1, 6, 6, 348, -570, 12084, -31332, 780792, -6111930, 65506884, -599418444, 6707736456, -69508986852, 738378468744, -7878832564872, 85524000547056, -929068361832378, 10158667075255524, -111690827626777788, 1234592278534799592, -13700571880245603276, 152613494540593338264
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 27 2024

Keywords

Crossrefs

Cf. A032308, A370714.
Cf. A300579, A344062.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 + 3*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 4^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 3*(4*x)^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Sqrt[QPochhammer[-3, x]/4], {x, 0, nmax}], x] * 4^Range[0, nmax]

Formula

G.f.: Product_{k>=1} (1 + 3*(4*x)^k)^(1/2).
a(n) ~ (-1)^(n+1) * c * 12^n / n^(3/2), where c = QPochhammer(-1/3)^(1/2) / (2*sqrt(Pi)) = 0.311283382185276347775502154581850436407169685238...

A370735 a(n) = 5^(2*n) * [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

Views

Author

Vaclav Kotesovec, Feb 28 2024

Keywords

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)).

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

Views

Author

Vaclav Kotesovec, Feb 28 2024

Keywords

Crossrefs

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

Programs

  • Mathematica
    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]

Formula

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)).

A370751 a(n) = 2^n * [x^n] Product_{k>=1} ((1 + 3*x^k)/(1 - 3*x^k))^(1/2).

Original entry on oeis.org

1, 6, 30, 204, 966, 5748, 29388, 169944, 886278, 5169732, 27794820, 162920616, 894445212, 5274022920, 29398573272, 174041671344, 980746798278, 5821525480164, 33071756442708, 196663513473672, 1124154722216244, 6693497121210648, 38448301937075112, 229149691659210192
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 29 2024

Keywords

Crossrefs

Cf. A303390, A370752.
Cf. A032308, A242587.
Cf. A370711, A370714.

Programs

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

Formula

G.f.: Product_{k>=1} ((1 + 3*(2*x)^k)/(1 - 3*(2*x)^k))^(1/2).
a(n) ~ c * 6^n / n^(1/2), where c = (QPochhammer(-1,1/3) / (Pi * QPochhammer(1/3)))^(1/2) = 1.333660169175690343841707335109800906849893636...
Showing 1-4 of 4 results.

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