A292037 -id:A292037 - 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)).

A294755 Expansion of Product_{k>=1} ((1 + x^(2k - 1))/(1 - x^(2k - 1)))^(k^2).

Original entry on oeis.org

1, 2, 2, 10, 18, 36, 86, 150, 326, 608, 1164, 2230, 4046, 7632, 13622, 24868, 44222, 78304, 138312, 240138, 418648, 718292, 1233494, 2097350, 3552370, 5987642, 10026088, 16745600, 27779030, 45970868, 75650248, 124100970, 202720814, 329909400, 535132036

Offset: 0

Vaclav Kotesovec, Nov 08 2017

nonn

Convolution of A294749 and A294750.

In general, if g.f. = Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k-1)))^(c2*k^2 + c1*k + c0) and c2 > 0, then a(n) ~ exp(Pi*sqrt(2) * c2^(1/4) * n^(3/4) / 3 + 7*(c1+c2) * Zeta(3) * sqrt(n) / (2*sqrt(c2) * Pi^2) + (Pi*(4*c0 + 2*c1 + c2) / (8*sqrt(2) * c2^(1/4)) - 49*(c1+c2)^2 * Zeta(3)^2 / (2^(3/2) * c2^(5/4) * Pi^5)) * n^(1/4) - (7*c0 + 21*c1/4 + c2 + 7*c0*c1/c2 + 7*c1^2/(2*c2)) * Zeta(3) / (4*Pi^2) + 22411*(c1+c2)^3 * Zeta(3)^3 / (196 * c2^2 * Pi^8) - (c1+c2)/24) * A^((c1+c2)/2) * (n^((c1+c2)/96 - 5/8) / (2^(c0/2 + (11*c1 + 5*c2)/48 + 9/4) * Pi^((c1+c2)/24) * c2^((c1+c2)/96 - 1/8))), where A is the Glaisher-Kinkelin constant A074962.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A206622, A292037, A294749, A294750.

nmax = 50; CoefficientList[Series[Product[((1+x^(2*k-1))/(1-x^(2*k-1)))^(k^2), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(sqrt(2)*Pi * n^(3/4)/3 + 7*Zeta(3) * sqrt(n) / (2*Pi^2) + (Pi / (8*sqrt(2)) - 49*Zeta(3)^2 / (2^(3/2) * Pi^5)) * n^(1/4) + 22411*Zeta(3)^3 / (196*Pi^8) - Zeta(3)/(4*Pi^2) - 1/24) * sqrt(A) / (2^(113/48) * Pi^(1/24) * n^(59/96)), where A is the Glaisher-Kinkelin constant A074962.

A295831 Expansion of Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^k.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 19, 30, 47, 76, 118, 181, 277, 417, 624, 929, 1367, 2001, 2913, 4210, 6056, 8665, 12328, 17466, 24640, 34600, 48395, 67442, 93625, 129520, 178588, 245429, 336252, 459324, 625613, 849762, 1151150, 1555378, 2096332, 2818630, 3780903, 5060240, 6757633, 9005106, 11975265

Offset: 0

Ilya Gutkovskiy, Nov 28 2017

nonn

Cf. A001935, A001936, A001937, A026007, A035528, A093160, A113415, A156616, A284474, A292037, A295832.

nmax = 44; CoefficientList[Series[Product[((1 + x^(2 k))/(1 - x^(2 k - 1)))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 44; CoefficientList[Series[Exp[Sum[x^k (1 - (-1)^k x^k)/(k (1 - x^(2 k))^2), {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^k.
G.f.: exp(Sum_{k>=1} x^k*(1 - (-1)^k*x^k)/(k*(1 - x^(2*k))^2)).
a(n) ~ exp(3*(7*Zeta(3))^(1/3) * n^(2/3) / 4 + Pi^2 * n^(1/3) / (12 * (7*Zeta(3))^(1/3)) - Pi^4 / (3024*Zeta(3)) - 1/24) * A^(1/2) * (7*Zeta(3))^(11/72) / (2^(11/8) * sqrt(3*Pi) * n^(47/72)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 28 2017

A294779 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k-1)))^(k*(k-1)/2).

Original entry on oeis.org

1, 0, 0, 2, 0, 6, 2, 12, 12, 22, 42, 42, 114, 102, 264, 280, 564, 744, 1186, 1866, 2538, 4380, 5598, 9732, 12602, 20898, 28374, 44048, 63000, 92190, 137012, 192864, 291588, 403668, 609072, 843228, 1253978, 1752150, 2555058, 3611380, 5168778, 7371324, 10400908

Offset: 0

Vaclav Kotesovec, Nov 08 2017

nonn

Convolution of A294777 and A294778.

Cf. A206622, A292037, A294755, A294777, A294778, A294780.

nmax = 50; CoefficientList[Series[Product[((1+x^(2*k-1))/(1-x^(2*k-1)))^(k*(k-1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi * 2^(1/4) * n^(3/4)/3 - Pi*n^(1/4) / 2^(17/4) + 3*Zeta(3) / (32*Pi^2)) / (2^(37/16) * n^(5/8)).

cp's OEIS Frontend

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

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A294755 Expansion of Product_{k>=1} ((1 + x^(2k - 1))/(1 - x^(2k - 1)))^(k^2).

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A295831 Expansion of Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^k.

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A294779 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k-1)))^(k*(k-1)/2).

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cp's OEIS Frontend

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

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A294755 Expansion of Product_{k>=1} ((1 + x^(2*k - 1))/(1 - x^(2*k - 1)))^(k^2).

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A295831 Expansion of Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^k.

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A294779 Expansion of Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k-1)))^(k*(k-1)/2).

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A294755 Expansion of Product_{k>=1} ((1 + x^(2k - 1))/(1 - x^(2k - 1)))^(k^2).

A295831 Expansion of Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^k.

A294779 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k-1)))^(k*(k-1)/2).