A263140 -id:A263140 - cp's OEIS frontend

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

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.

A263145 Expansion of Product_{k>=1} (1+x^(5*k-1))^k.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, 0, 4, 4, 0, 0, 1, 10, 5, 0, 0, 6, 16, 6, 0, 0, 14, 28, 7, 0, 3, 32, 40, 8, 0, 10, 63, 60, 9, 0, 33, 112, 80, 10, 3, 74, 187, 110, 11, 14, 161, 300, 140, 12, 46, 308, 455, 182, 14, 120, 568, 672, 224, 26, 283

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. A263146, A263147, A263148, A263141, A263140, A262878, A263138.

nmax = 100; CoefficientList[Series[Product[(1+x^(5k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(4*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^(4*j)/(1 - x^(5*j))^2).

a(n) ~ 2^(27/100) * 3^(2/3) * 5^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(32400*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(1/3) * 5^(2/3) * n^(1/3) / (900*Zeta(3)^(1/3)) + Zeta(3)^(1/3) * 3^(4/3) * 2^(2/3) * 5^(1/3) * n^(2/3) / 20) / (30 * sqrt(Pi) * n^(2/3)).

A263138 Expansion of Product_{k>=1} (1+x^(4*k-1))^k.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 3, 0, 0, 4, 4, 0, 1, 10, 5, 0, 6, 16, 6, 0, 14, 28, 7, 3, 32, 40, 8, 10, 63, 60, 9, 33, 112, 80, 13, 74, 187, 110, 25, 161, 300, 140, 58, 308, 455, 183, 133, 568, 672, 236, 297, 968, 963, 321, 609, 1609, 1344, 468, 1188, 2546

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. A263139, A263136, A263140, A262878, A263145.

nmax = 100; CoefficientList[Series[Product[(1+x^(4k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(3*j)/(1 - x^(4*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^(3*j)/(1 - x^(4*j))^2).

a(n) ~ 2^(59/96) * 3^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(20736*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(2/3) * n^(1/3) / (288*Zeta(3)^(1/3)) + Zeta(3)^(1/3) * 2^(-8/3) * 3^(4/3) * n^(2/3)) / (12 * sqrt(Pi) * n^(2/3)).

A263149 Expansion of Product_{k>=1} (1 + x^(2*k+1))^k.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 3, 2, 4, 4, 5, 10, 7, 16, 13, 28, 22, 40, 41, 63, 73, 90, 123, 143, 199, 214, 316, 343, 483, 532, 733, 848, 1099, 1305, 1644, 2029, 2448, 3067, 3657, 4643, 5443, 6892, 8107, 10224, 12031, 14974, 17798, 21941, 26190, 31867, 38381, 46300

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, A263140, A263150, A263199.

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^(3*j)/(1 - x^(2*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

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

A292037 Expansion of Product_{k>=1} ((1 + x^(2k-1)) / (1 - x^(2k-1)))^k.

Original entry on oeis.org

1, 2, 2, 6, 10, 16, 30, 46, 78, 124, 196, 306, 470, 724, 1086, 1644, 2438, 3608, 5304, 7734, 11232, 16196, 23270, 33206, 47250, 66846, 94232, 132280, 184966, 257720, 357768, 495090, 682702, 938760, 1286668, 1758708, 2397012, 3258340, 4417570, 5974204, 8059824

Offset: 0

Vaclav Kotesovec, Sep 08 2017

nonn

Convolution of A263140 and A035528 (with a(0)=1).

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

Cf. A035528, A080054, A263140, A292038.

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

a(n) ~ exp(-1/24 - Pi^4/(1344*Zeta(3)) + Pi^2 * n^(1/3) / (8*(7*Zeta(3))^(1/3)) + 3*(7*Zeta(3))^(1/3) * n^(2/3)/4) * A^(1/2) * (7*Zeta(3))^(11/72) / (2^(5/4) * sqrt(3*Pi) * n^(47/72)), where A is the Glaisher-Kinkelin constant A074962.

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

Original entry on oeis.org

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)).

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.

A295832 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^k.

Original entry on oeis.org

1, 1, 1, 3, 5, 8, 12, 20, 33, 50, 74, 114, 175, 257, 375, 555, 814, 1171, 1677, 2406, 3435, 4855, 6825, 9591, 13428, 18667, 25851, 35745, 49250, 67544, 92340, 125966, 171345, 232257, 313945, 423470, 569778, 764465, 1023231, 1366827, 1821756, 2422394, 3214318, 4257088, 5627086, 7422941

Offset: 0

Ilya Gutkovskiy, Nov 28 2017

nonn

Cf. A000219, A006950, A113415, A156616, A224364, A263140, A273225, A273226, A274621, A295831.

nmax = 45; CoefficientList[Series[Product[((1 + x^(2 k - 1))/(1 - x^(2 k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Exp[Sum[x^k ((-1)^(k + 1) + 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))/(1 - x^(2*k)))^k.
G.f.: exp(Sum_{k>=1} x^k*((-1)^(k+1) + 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) / (24 * (7*Zeta(3))^(1/3)) - Pi^4 / (12096 * Zeta(3)) + 1/12) * (7*Zeta(3))^(7/36) / (A * 2^(23/24) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 28 2017

A294777 Expansion of Product_{k>=1} (1 + x^(2k-1))^(k(k-1)/2).

Original entry on oeis.org

1, 0, 0, 1, 0, 3, 0, 6, 3, 10, 9, 15, 28, 24, 60, 47, 126, 99, 227, 225, 414, 498, 717, 1044, 1301, 2082, 2364, 3984, 4482, 7353, 8513, 13287, 16317, 23698, 30789, 42081, 57499, 74763, 105276, 133273, 190155, 238122, 338291, 425775, 596142, 759651, 1041498

Offset: 0

Vaclav Kotesovec, Nov 08 2017

nonn

Cf. A027998, A027999, A263140, A294749.

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

a(n) ~ exp(Pi*14^(1/4) * n^(3/4) / (3^(5/4) * 5^(1/4)) - Pi*5^(1/4) * n^(1/4) / (2^(17/4) * 3^(3/4) * 7^(1/4))) * 7^(1/8) / (2^(19/8) * 15^(1/8) * n^(5/8)).

A319106 Expansion of Product_{k>=1} (1 + x^k)^ceiling(k/2).

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 11, 17, 26, 40, 60, 88, 131, 190, 276, 398, 568, 806, 1142, 1603, 2242, 3124, 4328, 5973, 8214, 11249, 15349, 20879, 28297, 38235, 51513, 69190, 92674, 123811, 164961, 219248, 290705, 384537, 507515, 668376, 878339, 1151899, 1507679, 1969503, 2567976, 3342227

Offset: 0

Ilya Gutkovskiy, Sep 10 2018

nonn

Weigh transform of 1, 1, 2, 2, 3, 3, 4, 4, ... (A110654).

N. J. A. Sloane, Transforms

Cf. A003293, A006054, A026007, A110654, A263140.

a:=series(mul((1+x^k)^ceil(k/2),k=1..100),x=0,46): seq(coeff(a,x,n),n=0..45); # Paolo P. Lava, Apr 02 2019

nmax = 45; CoefficientList[Series[Product[(1 + x^k)^Ceiling[k/2], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Product[((1 + x^(2 k - 1))(1 + x^(2 k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d Ceiling[d/2], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 45}]

G.f.: Product_{k>=1} (1 + x^k)^A110654(k).
G.f.: Product_{k>=1} ((1 + x^(2*k-1))*(1 + x^(2*k)))^k.
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d*ceiling(d/2) ) * x^k/k).
a(n) ~ exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(5/3) + Pi^2 * n^(1/3) / (2^(10/3) * 3^(4/3) * Zeta(3)^(1/3)) - Pi^4 / (2^7 * 3^4 * Zeta(3))) * Zeta(3)^(1/6) / (2^(7/8) * 3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Sep 11 2018

cp's OEIS Frontend

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

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A263145 Expansion of Product_{k>=1} (1+x^(5*k-1))^k.

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A263138 Expansion of Product_{k>=1} (1+x^(4*k-1))^k.

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A263149 Expansion of Product_{k>=1} (1 + x^(2*k+1))^k.

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A292037 Expansion of Product_{k>=1} ((1 + x^(2*k-1)) / (1 - x^(2*k-1)))^k.

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

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

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A295832 Expansion of Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k)))^k.

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A294777 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(k*(k-1)/2).

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A292037 Expansion of Product_{k>=1} ((1 + x^(2k-1)) / (1 - x^(2k-1)))^k.

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

A295832 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^k.

A294777 Expansion of Product_{k>=1} (1 + x^(2k-1))^(k(k-1)/2).