A263149 -id:A263149 - cp's OEIS frontend

A263140 Expansion of Product_{k>=1} (1 + x^(2*k-1))^k.

1, 1, 0, 2, 2, 3, 4, 5, 10, 11, 16, 20, 31, 39, 50, 71, 93, 124, 154, 211, 271, 357, 449, 587, 762, 968, 1233, 1571, 2021, 2535, 3220, 4049, 5145, 6431, 8070, 10105, 12670, 15784, 19619, 24447, 30348, 37635, 46464, 57532, 70945, 87477, 107456, 132192, 162220

Offset: 0

Vaclav Kotesovec, Oct 10 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A035528, A263149, A263150, A263199, A262878, A263138, A263145, A292037.

nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^j/(1 - x^(2*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^j/(1 - x^(2*j))^2).

a(n) ~ exp(-Pi^4 / (5184*Zeta(3)) + Pi^2 * n^(1/3) / (8 * 3^(4/3) * Zeta(3)^(1/3)) + 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)/4) * Zeta(3)^(1/6) / (2^(23/24) * 3^(1/3)* sqrt(Pi) * n^(2/3)).

A263150 Expansion of Product_{k>=1} 1/(1 - x^(2*k+1))^k.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 1, 3, 2, 5, 6, 7, 11, 12, 21, 22, 34, 38, 59, 67, 95, 118, 155, 198, 252, 330, 409, 540, 662, 867, 1067, 1382, 1705, 2187, 2705, 3430, 4267, 5348, 6666, 8303, 10352, 12812, 15964, 19681, 24467, 30091, 37282, 45769, 56539, 69296, 85304

Offset: 0

Vaclav Kotesovec, Oct 10 2015

nonn

The side effect of this calculation is a formula: Integral_{x=0..infinity} exp(-3*x)/(x*(1-exp(-2*x))^2) - 1/(4*x^3) + 1/(4*x^2) - exp(-x)/(24*x) = log(2)/6 + log(A)/2 - 1/24, where A = A074962 is the Glaisher-Kinkelin constant.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A035528, A052847, A263140, A263149, A263199, A263352, A263395, A263396, A263397.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d-1, 2)=0, (d-1)/2, 0),
       d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # after Alois P. Heinz, Oct 17 2015

nmax = 100; CoefficientList[Series[Product[1/(1-x^(2*k+1))^k,{k,1,nmax}],{x,0,nmax}],x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(3*j)/(1 - x^(2*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} 1/j*x^(3*j)/(1 - x^(2*j))^2).

a(n) ~ sqrt(A) * Zeta(3)^(11/72) * exp(-1/24 - Pi^4/(1728*Zeta(3)) - Pi^2 * n^(1/3) / (3 * 2^(8/3)* Zeta(3)^(1/3)) + 3 * (Zeta(3)/2)^(1/3) * n^(2/3)/2) / (2^(35/72) * sqrt(3*Pi) * n^(47/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A263199 Expansion of Product_{k>=1} 1/(1 - x^(2k+1))^(2k+1).

Original entry on oeis.org

1, 0, 0, 3, 0, 5, 6, 7, 15, 19, 36, 41, 77, 100, 156, 230, 317, 482, 665, 981, 1354, 1967, 2710, 3852, 5363, 7453, 10373, 14287, 19780, 27022, 37220, 50583, 69140, 93693, 127098, 171640, 231469, 311323, 417627, 559577, 747122, 996947, 1325872, 1761900

Offset: 0

Vaclav Kotesovec, Oct 12 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A000219, A035528, A262811, A263140, A263149, A263150.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(d::even, 0, d), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
seq(b(n)-b(n-1), n=0..60);  # after Alois P. Heinz

nmax = 100; CoefficientList[Series[Product[1/(1 - x^(2*k+1))^(2*k+1),{k,1,nmax}],{x,0,nmax}],x]

For n>1, a(n) = A262811(n) - A262811(n-1).

a(n) ~ A * Zeta(3)^(17/36) * exp(-1/12 + 3 * Zeta(3)^(1/3) * n^(2/3)/2) / (2^(2/3) * sqrt(3*Pi) * n^(35/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A361008 G.f.: Product_{k >= 0} ((1 + x^(2k+1)) / (1 - x^(2k+1)))^k.

Original entry on oeis.org

1, 0, 0, 2, 0, 4, 2, 6, 8, 10, 20, 18, 42, 40, 78, 92, 140, 192, 258, 382, 480, 728, 902, 1334, 1698, 2404, 3148, 4292, 5742, 7608, 10304, 13430, 18192, 23592, 31720, 41144, 54766, 71188, 93762, 122156, 159420, 207820, 269380, 350726, 452434, 587520, 755446

Offset: 0

Vaclav Kotesovec, Apr 20 2023

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A080054, A291697, A263149, A263150.

Cf. A156616, A270919.

Table[SeriesCoefficient[Product[((1 + x^(2*k + 1))/(1 - x^(2*k + 1)))^k, {k, 0, n}], {x, 0, n}], {n, 0, 50}]

a(n) ~ sqrt(A/(3*Pi)) * (7*zeta(3))^(11/72) * exp(3*(7*zeta(3))^(1/3) * n^(2/3)/4 - Pi^2 * n^(1/3)/(8*(7*zeta(3))^(1/3)) - 1/24 - Pi^4/(1344*zeta(3))) / (2^(3/4) * n^(47/72)), where A = A074962 is the Glaisher-Kinkelin constant.

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A263140 Expansion of Product_{k>=1} (1 + x^(2*k-1))^k.

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A263150 Expansion of Product_{k>=1} 1/(1 - x^(2*k+1))^k.

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A263199 Expansion of Product_{k>=1} 1/(1 - x^(2k+1))^(2k+1).

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A361008 G.f.: Product_{k >= 0} ((1 + x^(2k+1)) / (1 - x^(2k+1)))^k.

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cp's OEIS Frontend

A263140 Expansion of Product_{k>=1} (1 + x^(2*k-1))^k.

Views

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A263150 Expansion of Product_{k>=1} 1/(1 - x^(2*k+1))^k.

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A263199 Expansion of Product_{k>=1} 1/(1 - x^(2*k+1))^(2*k+1).

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A361008 G.f.: Product_{k >= 0} ((1 + x^(2*k+1)) / (1 - x^(2*k+1)))^k.

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A263199 Expansion of Product_{k>=1} 1/(1 - x^(2k+1))^(2k+1).

A361008 G.f.: Product_{k >= 0} ((1 + x^(2k+1)) / (1 - x^(2k+1)))^k.