A294755 -id:A294755 - cp's OEIS frontend

A294749 Expansion of Product_{k>=1} (1 + x^(2*k - 1))^(k^2).

1, 1, 0, 4, 4, 9, 15, 22, 52, 65, 129, 190, 335, 534, 814, 1399, 2074, 3462, 5135, 8303, 12658, 19562, 30182, 45542, 70620, 105034, 161223, 239532, 362929, 539252, 805320, 1197589, 1769483, 2624604, 3847755, 5681787, 8291848, 12165978, 17696362, 25796820

Offset: 0

Vaclav Kotesovec, Nov 08 2017

nonn

In general, if g.f. = Product_{k>=1} (1 + x^(2*k-1))^(c2*k^2 + c1*k + c0) and c2>0, then a(n) ~ exp(Pi*sqrt(2)/3 * (7*c2/15)^(1/4) * n^(3/4) + 3*(c1+c2) * Zeta(3) / (2*Pi^2) * sqrt(15*n/(7*c2)) + (Pi*(4*c0 + 2*c1 + c2) * (15/(7*c2))^(1/4) / (24*sqrt(2)) - 9*(c1+c2)^2 * Zeta(3)^2 * (15/(7*c2))^(5/4) / (2^(3/2) * Pi^5)) * n^(1/4) + 2025*(c1+c2)^3 * Zeta(3)^3 / (49 * c2^2 * Pi^8) - 15*(c1+c2) * (4*c0 + 2*c1 + c2) * Zeta(3) / (112 * c2 * Pi^2)) * (7/15)^(1/8) * 2^((c1+c2)/24 - 9/4) * c2^(1/8) / n^(5/8).

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

Cf. A027998, A263140, A294750, A294755.

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

a(n) ~ exp(Pi/3 * (7/15)^(1/4) * sqrt(2) * n^(3/4) + 3*Zeta(3) * sqrt(15*n/7) / (2*Pi^2) + (Pi * (15/7)^(1/4) / (24*sqrt(2)) - 9*Zeta(3)^2 * (15/7)^(5/4) / (2^(3/2) * Pi^5)) * n^(1/4) + 2025*Zeta(3)^3 / (49*Pi^8) - 15*Zeta(3) / (112*Pi^2)) * (7/15)^(1/8) / (2^(53/24) * n^(5/8)).

A294750 Expansion of Product_{k>=1} 1/(1 - x^(2*k - 1))^(k^2).

Original entry on oeis.org

1, 1, 1, 5, 5, 14, 24, 40, 76, 121, 230, 356, 635, 1024, 1709, 2820, 4510, 7430, 11712, 19007, 29800, 47490, 74261, 116385, 181423, 280696, 434956, 666970, 1025816, 1562504, 2383916, 3611493, 5467505, 8241296, 12389888, 18581326, 27765501, 41426994, 61573390

Offset: 0

Vaclav Kotesovec, Nov 08 2017

nonn

In general, if g.f. = Product_{k>=1} 1/(1 - x^(2*k-1))^(c2*k^2 + c1*k + c0) and c2>0, then a(n) ~ exp(2*Pi/3 * (2*c2/15)^(1/4) * n^(3/4) + (c1+c2) * Zeta(3) / Pi^2 * sqrt(15*n/(2*c2)) + (Pi*(4*c0 + 2*c1 + c2)/24 - 15*(c1+c2)^2 * Zeta(3)^2 / (2*c2*Pi^5)) * (15*n/(2*c2))^(1/4) + 75*(c1+c2)^3 * Zeta(3)^3 / (c2^2 * Pi^8) - (5*c0 + 15*c1/4 + c2/2 + 5*c1*(2*c0 + c1) / (2*c2)) * (Zeta(3) / (4*Pi^2)) - (c1+c2)/24) * A^((c1+c2)/2) * (15/c2)^((c1+c2)/96 - 1/8) * n^((c1+c2)/96 - 5/8) / (2^(15/8 + c0/2 + (29*c1 + 17*c2)/96) * Pi^((c1+c2)/24)), where A is the Glaisher-Kinkelin constant A074962.

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

Cf. A023871, A035528, A294749, A294755.

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

a(n) ~ exp(2*Pi/3 * (2/15)^(1/4) * n^(3/4) + Zeta(3) * sqrt(15*n/2) / Pi^2 + (Pi * (15/2)^(1/4)/24 - Zeta(3)^2 * (15/2)^(5/4) / Pi^5) * n^(1/4) + 75*Zeta(3)^3 / Pi^8 - Zeta(3) / (8*Pi^2) - 1/24) * sqrt(A) / (2^(197/96) * 15^(11/96) * Pi^(1/24) * n^(59/96)), where A is the Glaisher-Kinkelin constant A074962.

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)).

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A294749 Expansion of Product_{k>=1} (1 + x^(2*k - 1))^(k^2).

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A294750 Expansion of Product_{k>=1} 1/(1 - x^(2*k - 1))^(k^2).

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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

A294749 Expansion of Product_{k>=1} (1 + x^(2*k - 1))^(k^2).

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A294750 Expansion of Product_{k>=1} 1/(1 - x^(2*k - 1))^(k^2).

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