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 1-3 of 3 results.

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

Original entry on oeis.org

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

A370737 a(n) = 5^(2*n) * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/5).

Original entry on oeis.org

1, 10, 50, 14750, -166250, 14011250, -133418750, 18136968750, -620089531250, 29520532031250, -917207280468750, 51260806902343750, -2257145499863281250, 101035630688769531250, -4434459153208496093750, 214279556679692871093750, -9859289197933918457031250, 454976266920750451660156250
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 28 2024

Keywords

Crossrefs

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

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1+2*x^k, {k, 1, nmax}]^(1/5), {x, 0, nmax}], x] * 25^Range[0, nmax]
nmax = 20; CoefficientList[Series[Product[1+2*(25*x)^k, {k, 1, nmax}]^(1/5), {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + 2*(25*x)^k)^(1/5).
a(n) ~ (-1)^(n+1) * QPochhammer(-1/2)^(1/5) * 50^n / (5 * Gamma(4/5) * n^(6/5)).

A370749 a(n) = 2^n * [x^n] Product_{k>=1} ((1 + 2*x^k)/(1 - 2*x^k))^(1/4).

Original entry on oeis.org

1, 2, 6, 28, 70, 300, 892, 3544, 9990, 43340, 127988, 546120, 1651356, 7227896, 22414008, 99344944, 312879302, 1396285452, 4486205892, 20057934312, 65293087284, 292353604136, 963327294536, 4308913730256, 14340603113372, 64059675491512, 215075154021384, 958968160741328
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 29 2024

Keywords

Crossrefs

Cf. A261584, A303346, A370750.
Cf. A032302, A070933.
Cf. A370736, A370732.

Programs

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

Formula

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

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