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.

A370765 a(n) = 9^n * [x^n] Product_{k>=1} ((1 + 2^(k+1)*x^k) * (1 + 2^(k-1)*x^k))^(1/3).

Original entry on oeis.org

1, 15, 153, 11295, 31968, 5289300, 41957514, 3216919050, -21009764691, 2153132775315, -16978376482767, 1659596014366335, -35929151338082922, 1473739361689662990, -38968782475183427016, 1541715187631618436300, -46858796372722560413526, 1615119529247884664988030
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 01 2024

Keywords

Crossrefs

Cf. A032302, A304961, A370016, A370761, A370764.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Product[(1+2^(k+1)*x^k)*(1+2^(k-1)*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 9^Range[0, nmax]
nmax = 25; CoefficientList[Series[Product[(1+2^(k+1)*(9*x)^k)*(1+2^(k-1)*(9*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[(2*QPochhammer[-2, 2*x]*QPochhammer[-1/2, 2*x]/9)^(1/3), {x, 0, nmax}], x] * 9^Range[0, nmax]
nmax = 25; CoefficientList[Series[(2*QPochhammer[-2, x]*QPochhammer[-1/2, x]/9)^(1/3), {x, 0, nmax}], x] * 18^Range[0, nmax]

Formula

G.f.: Product_{k>=1} ((1 + 2^(k+1)*(9*x)^k) * (1 + 2^(k-1)*(9*x)^k))^(1/3).
a(n) ~ (-1)^(n+1) * c * 36^n / n^(4/3), where c = 0.244280405759762854740979712556383125782589356973734984...

A370764 a(n) = 4^n * [x^n] Product_{k>=1} ((1 + 2^(k+1)*x^k) * (1 + 2^(k-1)*x^k))^(1/2).

Original entry on oeis.org

1, 10, 62, 1620, 6966, 157580, 1284012, 19189160, 73908774, 2233414620, 9656822916, 287668788120, -324007115716, 40151699854200, -199460032590312, 7130611518222160, -64971542557275642, 1292318115470489340, -15433712240157937260, 265667290368470451000, -3624776372747687578668
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 01 2024

Keywords

Comments

In general, if d > 1 and g.f. = Product_{k>=1} ((1 + d^(k+1)*x^k) * (1 + d^(k-1)*x^k))^(1/2), then a(n) ~ (-1)^(n+1) * QPochhammer(-1/d) * d^(2*n) / (2*sqrt((1 + 1/d)*Pi) * n^(3/2)).

Crossrefs

Cf. A032302, A304961, A370709, A370761, A370765.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Product[(1+2^(k+1)*x^k)*(1+2^(k-1)*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 4^Range[0, nmax]
nmax = 25; CoefficientList[Series[Product[(1+2^(3*k+1)*x^k)*(1+2^(3*k-1)*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[(2*QPochhammer[-2, x]*QPochhammer[-1/2, x])^(1/2)/3, {x, 0, nmax}], x] * 8^Range[0, nmax]

Formula

G.f.: Product_{k>=1} ((1 + 2^(3*k+1)*x^k) * (1 + 2^(3*k-1)*x^k))^(1/2).
a(n) ~ (-1)^(n+1) * c * 16^n / n^(3/2), where c = QPochhammer(-1/2) / sqrt(6*Pi) = 0.278865402428524528968820654198674...

A370792 Expansion of Product_{k>=1} (1 + 3^(k+1)*x^k) * (1 + 3^(k-1)*x^k).

Original entry on oeis.org

1, 10, 39, 390, 1521, 7830, 49518, 207360, 951102, 4264650, 22185657, 89579520, 401428224, 1676401110, 7172977275, 31972081050, 130330236546, 537393139200, 2213787635712, 8988968449530, 36073295687070, 150459195064320, 590262148332288, 2362876271009370, 9314694641056095
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 02 2024

Keywords

Comments

In general, if d >= 1 and g.f. = Product_{k>=1} (1 + d^(k+1)*x^k) * (1 + d^(k-1)*x^k), then a(n) ~ d^(n + 1/2) * exp(sqrt(2*n*(Pi^2/3 + log(d)^2))) * (Pi^2/3 + log(d)^2)^(1/4) / (2^(5/4) * sqrt(Pi) * (d+1) * n^(3/4)).

Crossrefs

Cf. A032308, A344062.
Cf. A022567 (d=1), A370761 (d=2).

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Product[(1+3^(k+1)*x^k)*(1+3^(k-1)*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ 3^(n + 1/2) * exp(sqrt(2*n*(Pi^2/3 + log(3)^2))) * (Pi^2/3 + log(3)^2)^(1/4) / (2^(13/4) * sqrt(Pi) * n^(3/4)).
Showing 1-3 of 3 results.

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