A262811 -id:A262811 - cp's OEIS frontend

A262736 Expansion of Product_{k>=1} (1 + x^(2k-1))^(2k-1).

1, 1, 0, 3, 3, 5, 8, 10, 22, 25, 41, 57, 88, 126, 168, 261, 351, 512, 685, 984, 1357, 1865, 2566, 3485, 4838, 6459, 8832, 11831, 16056, 21404, 28660, 38259, 50875, 67613, 89161, 118184, 155321, 204609, 267708, 351125, 458331, 597740, 777590, 1010020, 1310390

Offset: 0

Vaclav Kotesovec, Sep 29 2015

nonn

Alois P. Heinz, 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, p. 20.

Cf. A026007, A026011, A255528, A262811, A292038.

Cf. A262924, A285292, A285293.

nmax = 60; CoefficientList[Series[Product[(1 + x^(2*k-1))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(3^(4/3) * (Zeta(3))^(1/3) * n^(2/3) / 2^(5/3)) * Zeta(3)^(1/6) / (2^(3/4) * 3^(1/3) * sqrt(Pi) * n^(2/3)).

A285069 Expansion of Product_{k>=1} (1 - x^(2k-1))^(2k-1).

Original entry on oeis.org

1, -1, 0, -3, 3, -5, 8, -10, 22, -25, 41, -57, 88, -126, 168, -261, 351, -512, 685, -984, 1357, -1865, 2566, -3485, 4838, -6459, 8832, -11831, 16056, -21404, 28660, -38259, 50875, -67613, 89161, -118184, 155321, -204609, 267708, -351125, 458331, -597740

Offset: 0

Seiichi Manyama, Apr 09 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A050999, A262736, A262811.

CoefficientList[Series[Product[(1 - x^(2k-1))^(2k-1), {k, 50}], {x, 0, 50}], x] (* Indranil Ghosh, Apr 09 2017 *)

N=66; x='x+O('x^N); Vec(exp(-sum(k=1, N, sumdiv(k, d, d^2*(d%2))*x^k/k))) \\ Seiichi Manyama, Oct 31 2017

a(n) = (-1)^n * A262736(n).
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A050999(k)*a(n-k) for n > 0.
a(n) ~ (-1)^n * exp(3^(4/3) * (Zeta(3))^(1/3) * n^(2/3) / 2^(5/3)) * Zeta(3)^(1/6) / (2^(3/4) * 3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Nov 09 2017

A262946 Expansion of Product_{k>=1} 1/(1-x^(3k-1))^(3k-1).

Original entry on oeis.org

1, 0, 2, 0, 3, 5, 4, 10, 13, 15, 37, 31, 61, 87, 99, 178, 228, 286, 477, 552, 816, 1163, 1418, 2077, 2790, 3507, 5113, 6478, 8563, 11888, 15005, 20100, 27054, 34055, 46002, 59905, 76436, 102105, 130879, 168103, 221954, 281300, 363743, 472557, 597579, 772148

Offset: 0

Vaclav Kotesovec, Oct 05 2015

nonn

A262946(n)/A262947(n) ~ exp(3*(d1-d2)) * Gamma(1/3)^3 / (2*Pi)^(3/2), where d1 = A263030 and d2 = A263031. - Vaclav Kotesovec, Oct 08 2015

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
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
Vaclav Kotesovec, Graph - The asymptotic ratio (120000 terms)

Cf. A035386, A262811, A262876, A262883, A262923, A262947.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+3, 3, 'r')=2, 3*r-1, 0),
       d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Oct 05 2015

nmax=60; CoefficientList[Series[Product[1/((1-x^(3k-1))^(3k-1)),{k,1,nmax}],{x,0,nmax}],x]
nmax=60; CoefficientList[Series[E^Sum[1/j*x^(2*j)*(2+x^(3*j))/(1-x^(3*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (2*Zeta(3))^(5/36) * exp(3*d1 + (3/2)^(2/3) * Zeta(3)^(1/3) * n^(2/3)) / (3^(29/36) * Gamma(2/3) * n^(23/36)), where d1 = A263030 = Integral_{x=0..infinity} 1/x*(exp(-2*x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 1/(9*x) + exp(-x)/36) = -0.18870819197952853237641009864920797359211446726842922150941... . - Vaclav Kotesovec, Oct 08 2015

A284628 Expansion of Product_{k>=1} 1/(1+x^(2k-1))^(2k-1).

Original entry on oeis.org

1, -1, 1, -4, 4, -9, 15, -22, 37, -56, 92, -133, 210, -310, 466, -696, 1013, -1495, 2160, -3141, 4495, -6462, 9172, -13024, 18387, -25840, 36213, -50500, 70280, -97302, 134522, -185105, 254245, -347938, 475036, -646676, 878145, -1189468, 1607095, -2166672, 2913794

Offset: 0

Seiichi Manyama, Apr 15 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A255528, A262736, A262811.

nmax = 50; CoefficientList[Series[Product[1/(1+x^(2*k-1))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

a(n) = (-1)^n * A262811(n).

A281781 Expansion of Product_{k>=1} (1 - x^(2k))^(2k)/(1 - x^(2k-1))^(2k-1).

Original entry on oeis.org

1, 1, -1, 2, -1, -2, 6, -6, 3, -1, -1, 9, -18, 23, -27, 23, -1, -24, 49, -89, 121, -117, 96, -60, -18, 138, -275, 408, -525, 592, -566, 444, -181, -276, 854, -1485, 2154, -2765, 3157, -3131, 2571, -1468, -301, 2813, -5860, 9153, -12386, 15082, -16664, 16558, -14125

Offset: 0

Seiichi Manyama, Apr 14 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A262811, A281683.

nmax = 50; CoefficientList[Series[Product[(1 - x^(2*k))^(2*k)/(1 - x^(2*k-1))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 09 2017 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(2*k))^(4*k)/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 09 2017 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^k)^(4*k)*(1 - x^k)^(3*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 09 2017 *)

x='x+O('x^51); Vec(prod(k=1, 50, (1 - x^(2*k))^(2*k)/(1 - x^(2*k-1))^(2*k-1))) \\ Indranil Ghosh, Apr 14 2017

G.f.: exp(Sum_{k>=1} x^k/(k*(1 + x^k)^2)). - Ilya Gutkovskiy, May 28 2018

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.

A285048 Expansion of Product_{k>=0} 1/(1-x^(4k+1))^(4k+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 6, 6, 6, 15, 30, 30, 30, 43, 88, 123, 123, 140, 250, 385, 455, 476, 678, 1098, 1413, 1564, 1913, 2918, 4048, 4707, 5452, 7572, 10747, 13265, 15195, 19534, 27349, 35146, 41042, 50011, 67596, 88897, 106519, 126635, 164230, 216862, 266473, 314883

Offset: 0

Seiichi Manyama, Apr 15 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Vaclav Kotesovec)

Product_{k>=0} 1/(1-x^(m*k+1))^(m*k+1): A262811 (m=2), A262947 (m=3), this sequence (m=4), A285049 (m=5).

Cf. A285070, A285287.

nmax = 50; CoefficientList[Series[Product[1/(1-x^(4*k-3))^(4*k-3), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 16 2017 *)

a(n) ~ 4 * Pi * 2^(25/72) * Zeta(3)^(11/72) * exp(4*c + 3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3)) / (sqrt(3) * Gamma(1/4)^3 * n^(47/72)), where c = Integral_{x=0..inf} ((-19/(exp(x)*96) + 1/(exp(x)*(1 - exp(-4*x))^2) - 1/(16*x^2) - 3/(16*x))/x) dx = 0.09601010361866957956805888476415949391295401812706635... - Vaclav Kotesovec, Apr 16 2017

A285131 Expansion of Product_{k>=0} 1/(1-x^(4k+3))^(4k+3).

Original entry on oeis.org

1, 0, 0, 3, 0, 0, 6, 7, 0, 10, 21, 11, 15, 42, 61, 36, 70, 150, 150, 124, 278, 441, 375, 468, 909, 1131, 1018, 1581, 2602, 2810, 2947, 4819, 6768, 6980, 8509, 13389, 16788, 17609, 23722, 34720, 40337, 44863, 63128, 85430, 95887, 114037, 159882, 202699, 227087

Offset: 0

Seiichi Manyama, Apr 15 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k>=0} 1/(1-x^(m*k+m-1))^(m*k+m-1): A262811 (m=2), A262946 (m=3), this sequence (m=4), A285132 (m=5).

Cf. A285213.

nmax = 100; CoefficientList[Series[Product[1/(1-x^(4*k+3))^(4*k+3), {k,0,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

x='x+O('x^100); Vec(prod(k=0, 100, 1/(1 - x^(4*k + 3))^(4*k + 3))) \\ Indranil Ghosh, Apr 15 2017

a(n) ~ exp(4*c + 3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(11/72) / (2^(47/72) * sqrt(3) * Gamma(3/4) * n^(47/72)), where c = Integral_{x=0..inf} ((5/(exp(x)*96) + 1/(exp(3*x)*(1 - exp(-4*x))^2) - 1/(16*x^2) - 1/(16*x))/x) dx = -0.158924147180165035059952001737321408554746599955833696821824808... - Vaclav Kotesovec, Apr 15 2017

A285215 Expansion of Product_{k>=1} (1 - x^(4k))^(4k) / (1 - x^k)^k.

Original entry on oeis.org

1, 1, 3, 6, 9, 20, 36, 62, 106, 184, 302, 503, 829, 1325, 2119, 3367, 5282, 8227, 12740, 19550, 29849, 45300, 68325, 102495, 152998, 227249, 336005, 494597, 724875, 1058213, 1538860, 2229370, 3218304, 4630015, 6638728, 9488894, 13520995, 19208916, 27211430

Offset: 0

Seiichi Manyama, Apr 15 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k>=1} (1 - x^(m*k))^(m*k)/(1 - x^k)^k: A262811 (m=2), A262923 (m=3), this sequence (m=4), A285246 (m=5).

Cf. A285262, A285284.

nmax = 50; CoefficientList[Series[Product[1 / ((1-x^(4*k+1))^(4*k+1) * (1-x^(4*k+2))^(4*k+2) * (1-x^(4*k+3))^(4*k+3)), {k,0,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(4*k))^(4*k)/((1 - x^k)^k), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

x='x+O('x^100); Vec(prod(k=0, 100, 1 / ((1 - x^(4*k + 1))^(4*k + 1)*(1 - x^(4*k + 2))^(4*k + 2)*(1 - x^(4*k + 3))^(4*k + 3)))) \\ Indranil Ghosh, Apr 15 2017

G.f.: Product_{k>=0} 1 / ((1-x^(4*k+1))^(4*k+1) * (1-x^(4*k+2))^(4*k+2) * (1-x^(4*k+3))^(4*k+3)).

a(n) ~ exp(-1/4 + 2^(-4/3) * 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)) * A^3 * Zeta(3)^(1/12) / (2^(5/4) * 3^(5/12) * sqrt(Pi) * n^(7/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 16 2017

A292038 Expansion of Product_{k>=1} ((1 + x^(2k-1)) / (1 - x^(2k-1)))^(2*k-1).

Original entry on oeis.org

1, 2, 2, 8, 14, 24, 52, 84, 158, 274, 464, 800, 1316, 2208, 3576, 5832, 9358, 14876, 23614, 36936, 57752, 89336, 137716, 210844, 321148, 486890, 733912, 1102336, 1646736, 2451464, 3632832, 5363988, 7889710, 11562712, 16888748, 24581904, 35670242, 51591096

Offset: 0

Vaclav Kotesovec, Sep 08 2017

nonn

Convolution of A262736 and A262811.

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

Cf. A080054, A262736, A262811, A284628, A285069, A292038.

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

a(n) ~ exp(3*(7*Zeta(3))^(1/3)*n^(2/3) / 2^(5/3) - 1/12) * A * (7*Zeta(3))^(5/36) / (2^(31/36) * sqrt(3*Pi) * n^(23/36)), where A is the Glaisher-Kinkelin constant A074962.

G.f.: exp(2*Sum_{k>=1} sigma_2(2*k - 1)*x^(2*k-1)/(2*k - 1)). - Ilya Gutkovskiy, Apr 19 2019

cp's OEIS Frontend

A262736 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(2*k-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A285069 Expansion of Product_{k>=1} (1 - x^(2*k-1))^(2*k-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A262946 Expansion of Product_{k>=1} 1/(1-x^(3*k-1))^(3*k-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A284628 Expansion of Product_{k>=1} 1/(1+x^(2*k-1))^(2*k-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A281781 Expansion of Product_{k>=1} (1 - x^(2*k))^(2*k)/(1 - x^(2*k-1))^(2*k-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A285048 Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^(4*k+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A285131 Expansion of Product_{k>=0} 1/(1-x^(4*k+3))^(4*k+3).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A285215 Expansion of Product_{k>=1} (1 - x^(4*k))^(4*k) / (1 - x^k)^k.

A262736 Expansion of Product_{k>=1} (1 + x^(2k-1))^(2k-1).

A285069 Expansion of Product_{k>=1} (1 - x^(2k-1))^(2k-1).

A262946 Expansion of Product_{k>=1} 1/(1-x^(3k-1))^(3k-1).

A284628 Expansion of Product_{k>=1} 1/(1+x^(2k-1))^(2k-1).

A281781 Expansion of Product_{k>=1} (1 - x^(2k))^(2k)/(1 - x^(2k-1))^(2k-1).

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

A285048 Expansion of Product_{k>=0} 1/(1-x^(4k+1))^(4k+1).

A285131 Expansion of Product_{k>=0} 1/(1-x^(4k+3))^(4k+3).

A285215 Expansion of Product_{k>=1} (1 - x^(4k))^(4k) / (1 - x^k)^k.

A292038 Expansion of Product_{k>=1} ((1 + x^(2k-1)) / (1 - x^(2k-1)))^(2*k-1).