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 1-7 of 7 results.

A298994 Expansion of Product_{n>=1} (1 + (4*x)^n)^(1/2).

Original entry on oeis.org

1, 2, 6, 52, 134, 956, 4124, 20008, 73158, 439660, 1874612, 8350808, 37583004, 169862616, 779948152, 3774085968, 15435601222, 69542934604, 329825707332, 1403190752632, 6313190864052, 29079505547912, 126937389732872, 552273916408368, 2477249228318748
Offset: 0

Views

Author

Seiichi Manyama, Jan 31 2018

Keywords

Links

Crossrefs

Cf. A271235, A298411, A298993, A303074, A370739.

Programs

  • Mathematica
    CoefficientList[Series[Sqrt[QPochhammer[-1, 4*x]/2], {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 18 2018 *)

Formula

Convolution inverse of A298993.
a(n) ~ 2^(2*n - 2) * exp(Pi*sqrt(n/6)) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 18 2018
Sum_{k=0..n} a(k)*a(n-k) = 4^n * A000009(n). - Vaclav Kotesovec, Jun 07 2025

A271236 G.f.: Product_{k>=1} 1/(1 - (9*x)^k)^(1/3).

Original entry on oeis.org

1, 3, 45, 450, 5805, 52326, 705591, 6190425, 77219325, 751178610, 8522919063, 80502824835, 975122402985, 8949951461925, 100088881882830, 1003346683458480, 10828622925516312, 104307212166072165, 1152197107898173875, 11048535008792967825, 119509353627934830327
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 02 2016

Keywords

Comments

In general, for h>=1, if g.f. = Product_{k>=1} 1/(1-(h^2*x)^k)^(1/h), then a(n) ~ h^(2*n) * exp(Pi*sqrt(2*n/(3*h))) / (2^(5*h+3) * 3^(h+1) * h^(h+1) * n^(3*h+1))^(1/(4*h)).
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/3, g(n) = 9^n. - Seiichi Manyama, Apr 20 2018

Links

Crossrefs

Cf. A298994, A303074, A303130, A303152, A303342, A370735.
Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(-1/b): A000041 (b=1), A271235 (b=2), this sequence (b=3), A303135 (b=4), A303136 (b=5).

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[1/(1 - (9*x)^k)^(1/3), {k, 1, nmax}], {x, 0, nmax}], x]
  • PARI
    N=99; x='x+O('x^N); Vec(prod(k=1, N, 1/(1-(9*x)^k)^(1/3))) \\ Altug Alkan, Apr 20 2018

Formula

a(n) ~ 3^(2*n - 2/3) * exp(sqrt(2*n)*Pi/3) / (2^(3/2) * n^(5/6)).

A298411 Coefficients of q^(-1/24)*eta(4q)^(1/2).

Original entry on oeis.org

1, -2, -10, -20, -90, 132, -836, 6040, 2310, 60180, 180308, 1662568, -2995620, 24401320, 44072120, -102437328, 19390406, 2649221300, -10584460060, 14475802440, -228570333836, -815899620616, 2088529753800, -5590702681520, -100828534100580, -172013432412024
Offset: 0

Views

Author

William J. Keith, Jan 18 2018

Keywords

Comments

The q^(kn) term of any single factor of the product (1-(4q)^k)^(1/2) is (-2)*A000108(n-1). Hence these numbers are related to the Catalan numbers A000108 by a partition-based convolution.
Sequence appears to be positive and negative roughly half the time.
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/2, g(n) = 4^n. - Seiichi Manyama, Apr 20 2018

Links

Crossrefs

Cf. A000108, A271235, A298994.
Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(1/b): A010815 (b=1), this sequence (b=2), A303152 (b=3), A303153 (b=4), A303154 (b=5).

Programs

  • Mathematica
    Series[Product[(1 - (4 q)^k)^(1/2), {k, 1, 100}], {q, 0, 100}]
  • PARI
    q='q+O('q^99); Vec(eta(4*q)^(1/2)) \\ Altug Alkan, Apr 20 2018

Formula

G.f.: Product_{k>=1} (1 - (4x)^k)^(1/2).

A298993 Expansion of Product_{n>=1} 1/sqrt(1 + (4*x)^n).

Original entry on oeis.org

1, -2, -2, -36, 54, -476, 556, -6088, 35878, -156844, 444164, -1734648, 11948604, -35313048, 156354328, -864527760, 4733447686, -12692853452, 54065039380, -226098757912, 1278838329812, -5257771138376, 19455009120232, -76455773381360, 453306681446748
Offset: 0

Views

Author

Seiichi Manyama, Jan 31 2018

Keywords

Links

Crossrefs

Cf. A000984, A081362, A271235, A298994.

Programs

  • Mathematica
    CoefficientList[Series[Sqrt[2/QPochhammer[-1, 4*x]], {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 18 2018 *)

Formula

a(n) ~ (-1)^n * exp(Pi*sqrt(n/12)) * 4^(n-1) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 18 2018

A303135 Expansion of Product_{n>=1} (1 - (16*x)^n)^(-1/4).

Original entry on oeis.org

1, 4, 104, 1760, 39520, 590720, 14285056, 205151232, 4596467200, 75375073280, 1504196046848, 23673049726976, 525315968712704, 7912159583600640, 158055039529779200, 2726833423421800448, 51889395654107463680, 840470097284214292480, 16765991910040314839040
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/4, g(n) = 16^n.

Links

Crossrefs

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(-1/b): A000041 (b=1), A271235 (b=2), A271236 (b=3), this sequence (b=4), A303136 (b=5).
Cf. A303131, A303153.

Programs

  • Mathematica
    CoefficientList[Series[1/QPochhammer[16*x]^(1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 19 2018 *)

Formula

a(n) ~ exp(sqrt(n/6)*Pi) * 2^(4*n - 33/16) / (3^(5/16) * n^(13/16)). - Vaclav Kotesovec, Apr 19 2018

A303136 Expansion of Product_{n>=1} (1 - (25*x)^n)^(-1/5).

Original entry on oeis.org

1, 5, 200, 5125, 177500, 3952500, 150715625, 3185187500, 112844843750, 2783033593750, 86330708203125, 2019237027343750, 72195817812500000, 1591910699609375000, 50158322275878906250, 1322261581989501953125, 39183430287559814453125, 946961406814801025390625
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2018

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/5, g(n) = 25^n.

Links

Crossrefs

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(-1/b): A000041 (b=1), A271235 (b=2), A271236 (b=3), A303135 (b=4), this sequence (b=5).
Cf. A303132, A303154.

Programs

  • Mathematica
    CoefficientList[Series[1/QPochhammer[25*x]^(1/5), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 19 2018 *)
CoefficientList[Series[Product[(1-(25x)^n)^(-1/5),{n,20}],{x,0,20}],x] (* Harvey P. Dale, Nov 04 2021 *)

Formula

a(n) ~ exp(Pi*sqrt(2*n/15)) * 5^(2*n - 3/10) / (2^(7/5) * 3^(3/10) * n^(4/5)). - Vaclav Kotesovec, Apr 19 2018

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)).
Showing 1-7 of 7 results.

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