A035528 -id:A035528 - cp's OEIS frontend

A263352 Expansion of Product_{k>=1} 1/(1 - x^(2*k+3))^k.

1, 0, 0, 0, 0, 1, 0, 2, 0, 3, 1, 4, 2, 5, 6, 7, 10, 9, 19, 14, 29, 23, 46, 38, 66, 64, 99, 107, 143, 171, 211, 270, 311, 418, 465, 633, 698, 945, 1049, 1399, 1579, 2052, 2364, 2997, 3527, 4366, 5219, 6339, 7686, 9197, 11234, 13321, 16340, 19261, 23622, 27796

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

From Vaclav Kotesovec, Oct 17 2015: (Start)
In general, if g.f. = Product_{k>=1} 1/(1-x^(2*k+v))^k and v>0 is odd, then a(n) ~ d2(v) * (2*n)^(v^2/24 - 25/36) * exp(-Pi^4 * v^2 / (1728*Zeta(3)) - Pi^2 * v * n^(1/3) /(3 * 2^(8/3) * Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) / (sqrt(3*Pi) * Zeta(3)^(v^2/24 - 7/36)), where Zeta(3) = A002117.
d2(v) = exp(Integral_{x=0..infinity} 1/(x*exp((v-2)*x) * (exp(2*x)- 1)^2) - (3*v^2-2)/(24*x*exp(x)) + v/(4*x^2) - 1/(4*x^3) dx).
d2(v) = 2^(v/4 - 1/12) * exp(-Zeta'(-1)/2) / Product_{j=1..(v-1)/2} (2*j-1)!!, where Zeta'(-1) = A084448 and Product_{j=1..(v-1)/2} (2*j-1)!! = A057863((v-1)/2).
d2(v) = 2^(1/12 + v/4 - v^2/8) * exp(1/12) * Pi^(v/4) / (A * G(v/2 + 1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
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
Eric Weisstein's World of Mathematics, Barnes G-Function.

Cf. A052847, A000219, A263150, A035528, A263395, A263396, A263397.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(d>1 and d::odd, (d-3)/2, 0),
      d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 17 2015

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

a(n) ~ 2^(25/72) * sqrt(A) * exp(-1/24 + 3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3) - Pi^4/(192*Zeta(3)) - Pi^2 * n^(1/3)/(2^(8/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * Zeta(3)^(13/72) * n^(23/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

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)).

A263395 Expansion of Product_{k>=1} 1/(1 - x^(2*k+5))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 3, 0, 4, 1, 5, 2, 6, 6, 7, 10, 9, 19, 11, 28, 16, 44, 25, 61, 40, 87, 65, 116, 107, 160, 168, 215, 260, 295, 393, 407, 578, 573, 836, 814, 1193, 1167, 1675, 1684, 2335, 2427, 3238, 3501, 4468, 5014, 6161, 7152, 8494, 10121

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

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

Cf. A035528, A263150, A263352, A263396, A263397.

with(numtheory):
a:= proc(n) option remember; local r; `if`(n=0, 1,
       add(add(`if`(irem(d-4, 2, 'r')=1, d*r, 0)
       , d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 17 2015

nmax = 60; CoefficientList[Series[Product[1/(1 - x^(2*k+5))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(7*k)/(k*(1-x^(2*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^(7*k)/(k*(1-x^(2*k))^2)).

a(n) ~ 2^(109/72) * exp(-1/24 - 25*Pi^4/(1728*Zeta(3)) - 5*Pi^2 * n^(1/3) / (12 * 2^(2/3) * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) * sqrt(A) * n^(25/72) / (3*sqrt(3*Pi) * Zeta(3)^(61/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A263396 Expansion of Product_{k>=1} 1/(1 - x^(2*k+7))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 1, 6, 2, 7, 6, 8, 10, 9, 19, 11, 28, 13, 44, 18, 60, 27, 85, 42, 111, 67, 148, 109, 188, 169, 245, 260, 313, 390, 408, 568, 535, 811, 717, 1139, 974, 1568, 1343, 2134, 1872, 2873, 2621, 3832, 3687, 5088

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

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

Cf. A035528, A263150, A263352, A263395, A263397.

with(numtheory):
a:= proc(n) option remember; local r; `if`(n=0, 1,
       add(add(`if`(irem(d-6, 2, 'r')=1, d*r, 0)
       , d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..65);  # Alois P. Heinz, Oct 17 2015

nmax = 60; CoefficientList[Series[Product[1/(1 - x^(2*k+7))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(9*k)/(k*(1-x^(2*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^(9*k)/(k*(1-x^(2*k))^2)).

a(n) ~ 8 * 2^(1/72) * exp(-1/24 - 49*Pi^4/(1728*Zeta(3)) - 7*Pi^2 * n^(1/3) / (12 * 2^(2/3) * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) * sqrt(A) * n^(97/72) / (45*sqrt(3*Pi) * Zeta(3)^(133/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A263397 Expansion of Product_{k>=1} 1/(1 - x^(2*k+9))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 1, 7, 2, 8, 6, 9, 10, 10, 19, 11, 28, 13, 44, 15, 60, 20, 85, 29, 110, 44, 146, 69, 183, 111, 233, 171, 286, 262, 358, 391, 441, 568, 553, 808, 697, 1129, 898, 1543, 1174, 2080, 1563, 2766

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

In general, if g.f. = Product_{k>=1} 1/(1-x^(2*k+v))^k and v>0 is odd, then a(n) ~ d2(v) * (2*n)^(v^2/24 - 25/36) * exp(-Pi^4 * v^2 / (1728*Zeta(3)) - Pi^2 * v * n^(1/3) /(3 * 2^(8/3) * Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) / (sqrt(3*Pi) * Zeta(3)^(v^2/24 - 7/36)), where Zeta(3) = A002117.
d2(v) = exp(Integral_{x=0..infinity} 1/(x*exp((v-2)*x) * (exp(2*x)- 1)^2) - (3*v^2-2)/(24*x*exp(x)) + v/(4*x^2) - 1/(4*x^3) dx).
d2(v) = 2^(v/4 - 1/12) * exp(-Zeta'(-1)/2) / Product_{j=1..(v-1)/2} (2*j-1)!!, where Zeta'(-1) = A084448 and Product_{j=1..(v-1)/2} (2*j-1)!! = A057863((v-1)/2).
d2(v) = 2^(1/12 + v/4 - v^2/8) * exp(1/12) * Pi^(v/4) / (A * G(v/2 + 1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.

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
Eric Weisstein's World of Mathematics, Barnes G-Function.

Cf. A035528 (v=-1), A263150 (v=1), A263352 (v=3), A263395 (v=5), A263396 (v=7).

with(numtheory):
a:= proc(n) option remember; local r; `if`(n=0, 1,
       add(add(`if`(irem(d-8, 2, 'r')=1, d*r, 0)
       , d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..65);  # Alois P. Heinz, Oct 17 2015

nmax = 60; CoefficientList[Series[Product[1/(1 - x^(2*k+9))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(11*k)/(k*(1-x^(2*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^(11*k)/(k*(1-x^(2*k))^2)).

a(n) ~ 16 * 2^(61/72) * exp(-1/24 - 3*Pi^4/(64*Zeta(3)) - 3*Pi^2 * n^(1/3) / (2^(8/3) * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) * sqrt(A) * n^(193/72) / (4725*sqrt(3*Pi) * Zeta(3)^(229/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

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.

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.

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.

A294669 Expansion of Product_{k>=1} 1/(1 - x^(2k-1))^(k(3*k-1)/2).

Original entry on oeis.org

1, 1, 1, 6, 6, 18, 33, 55, 115, 185, 373, 604, 1113, 1903, 3251, 5678, 9350, 16153, 26420, 44561, 72912, 120150, 196329, 317988, 516881, 827778, 1333570, 2120492, 3381947, 5347513, 8447482, 13285450, 20813814, 32547272, 50638328, 78707858, 121738479

Offset: 0

Vaclav Kotesovec, Nov 06 2017

nonn

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

Cf. A035528, A262811, A294591, A278768.

nmax = 50; CoefficientList[Series[Product[1/(1-x^(2*k-1))^(k*(3*k-1)/2),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ exp(2*Pi * n^(3/4) / (3*5^(1/4)) + Zeta(3) * sqrt(5*n) / Pi^2 + 5^(1/4) * (Pi/48 - 5*Zeta(3)^2 / Pi^5) * n^(1/4) + 100*Zeta(3)^3 / (3*Pi^8) + 17*Zeta(3) / (96*Pi^2) - 1/24) * sqrt(A) / (2^(101/48) * 5^(11/96) * 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

cp's OEIS Frontend

A263352 Expansion of Product_{k>=1} 1/(1 - x^(2*k+3))^k.

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

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A263395 Expansion of Product_{k>=1} 1/(1 - x^(2*k+5))^k.

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A263396 Expansion of Product_{k>=1} 1/(1 - x^(2*k+7))^k.

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A263397 Expansion of Product_{k>=1} 1/(1 - x^(2*k+9))^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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A294750 Expansion of Product_{k>=1} 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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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).

A294669 Expansion of Product_{k>=1} 1/(1 - x^(2k-1))^(k(3*k-1)/2).

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