A262923 -id:A262923 - cp's OEIS frontend

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

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

A262947 Expansion of Product_{k>=1} 1/(1-x^(3k-2))^(3k-2).

Original entry on oeis.org

1, 1, 1, 1, 5, 5, 5, 12, 22, 22, 32, 60, 80, 93, 161, 231, 282, 404, 616, 775, 1041, 1535, 2037, 2600, 3708, 5029, 6411, 8710, 11968, 15315, 20189, 27444, 35619, 45939, 61605, 80422, 102932, 135481, 177391, 226263, 293561, 382984, 488826, 626558, 812750

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. A035382, A262877, A262883, A262923, A262946.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+3, 3, 'r')=1, 3*r-2, 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-2))^(3k-2)),{k,1,nmax}],{x,0,nmax}],x]
nmax=60; CoefficientList[Series[E^Sum[1/j*x^j*(1+2*x^(3*j))/(1-x^(3*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 2^(23/36) * sqrt(Pi) * Zeta(3)^(5/36) * exp(3*d2 + (3/2)^(2/3) * Zeta(3)^(1/3) * n^(2/3)) / (3^(11/36) * Gamma(1/3)^2 * n^(23/36)), where d2 = A263031 = Integral_{x=0..infinity} 1/x*(exp(-x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 2/(9*x) - 5*exp(-x)/36) = -0.01453742918328403360502029450226209036054149... . - Vaclav Kotesovec, Oct 08 2015

A262924 Expansion of Product_{k>=1} (1 + x^(3k-1))^(3k-1) * (1 + x^(3k-2))^(3k-2).

Original entry on oeis.org

1, 1, 2, 2, 5, 10, 13, 25, 35, 57, 87, 134, 211, 306, 458, 684, 996, 1465, 2129, 3073, 4411, 6288, 8977, 12707, 17913, 25185, 35231, 49078, 68228, 94490, 130408, 179425, 246121, 336681, 459239, 624842, 847986, 1147728, 1549773, 2087972, 2806455, 3764136

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

Convolution of A262948 and A262949.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
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. A262884, A262878, A262879, A262923, A261612, A262928, A262948, A262949.

Cf. A262736, A285292, A285293.

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

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

A262883 Expansion of Product_{k>=1} 1/((1-x^(3k-1))(1-x^(3*k-2)))^k.

Original entry on oeis.org

1, 1, 2, 2, 5, 7, 10, 15, 24, 33, 49, 68, 100, 136, 193, 267, 370, 501, 690, 928, 1260, 1687, 2265, 3007, 4006, 5289, 6987, 9163, 12033, 15698, 20469, 26572, 34470, 44510, 57442, 73861, 94852, 121439, 155287, 198007, 252165, 320335, 406396, 514410, 650288

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

Convolution of A262876 and A262877.

Vaclav Kotesovec and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 2001 terms from Vaclav Kotesovec)
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. A262876, A262877, A262884, A262923.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+3, 3, 'r')=0, 0, r), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 05 2015

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

a(n) ~ exp(-1/18 - Pi^4/(864*Zeta(3)) + (3*Zeta(3)/2)^(1/3) * n^(2/3) + Pi^2 * n^(1/3) / (2^(5/3)*3^(4/3)*Zeta(3)^(1/3))) * A^(2/3) * Gamma(4/3)^(1/3) * Zeta(3)^(7/54) / (2^(11/27) * 3^(79/108) * Pi^(2/3) * n^(17/27)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

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

A285246 Expansion of Product_{k>=1} (1 - x^(5k))^(5k) / (1 - x^k)^k.

Original entry on oeis.org

1, 1, 3, 6, 13, 19, 43, 71, 130, 217, 380, 619, 1049, 1685, 2757, 4404, 7027, 11014, 17326, 26820, 41488, 63514, 96970, 146808, 221659, 332212, 496439, 737535, 1091938, 1608564, 2361929, 3452736, 5031138, 7302373, 10566038, 15234196, 21900182, 31380435

Offset: 0

Seiichi Manyama, Apr 15 2017

nonn

In general, if m > 1 and g.f. = Product_{k>=1} (1 - x^(m*k))^(m*k)/((1 - x^k)^k), then a(n, m) ~ exp(1/12 - m/12 + 3 * 2^(-2/3) * (1-1/m)^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(-(m+11)/36) * A^(m-1) * (m-1)^((7-m)/36) * m^(-(2*m+7)/36) * Zeta(3)^((7-m)/36) * n^((m-25)/36) / sqrt(3*Pi), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 16 2017

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), A285215 (m=4), this sequence (m=5).

Cf. A285263, A285285.

nmax = 50; CoefficientList[Series[Product[1 / ((1-x^(5*k+1))^(5*k+1) * (1-x^(5*k+2))^(5*k+2) * (1-x^(5*k+3))^(5*k+3) * (1-x^(5*k+4))^(5*k+4)), {k,0,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^(5*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^(5*k + 1))^(5*k + 1)*(1 - x^(5*k + 2))^(5*k + 2)*(1 - x^(5*k + 3))^(5*k + 3)*(1 - x^(5*k + 4))^(5*k + 4)))) \\ Indranil Ghosh, Apr 15 2017

G.f.: Product_{k>=0} 1 / ((1-x^(5*k+1))^(5*k+1) * (1-x^(5*k+2))^(5*k+2) * (1-x^(5*k+3))^(5*k+3) * (1-x^(5*k+4))^(5*k+4)).

a(n) ~ exp(-1/3 + 3*(Zeta(3)/5)^(1/3)*n^(2/3)) * A^4 * Zeta(3)^(1/18) / (2^(1/3) * 5^(17/36) * sqrt(3*Pi) * n^(5/9)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 16 2017

A285247 Expansion of Product_{k>=1} (1-x^(3k-1))^(3k-1) * (1-x^(3k-2))^(3k-2).

Original entry on oeis.org

1, -1, -2, 2, -3, -2, 13, -5, -9, 35, -25, -34, 91, -78, -102, 240, -192, -233, 665, -441, -553, 1636, -1063, -1327, 3869, -2565, -3229, 8738, -6032, -7446, 19568, -13469, -16499, 43083, -29101, -35282, 93458, -61544, -74539, 198072, -128917, -155580, 412116, -267021

Offset: 0

Seiichi Manyama, Apr 15 2017

sign

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

Cf. A262923, A285050, A285212.

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

Convolution inverse of A262923.

cp's OEIS Frontend

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

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A262947 Expansion of Product_{k>=1} 1/(1-x^(3*k-2))^(3*k-2).

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A262924 Expansion of Product_{k>=1} (1 + x^(3*k-1))^(3*k-1) * (1 + x^(3*k-2))^(3*k-2).

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A262883 Expansion of Product_{k>=1} 1/((1-x^(3*k-1))*(1-x^(3*k-2)))^k.

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

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A285246 Expansion of Product_{k>=1} (1 - x^(5*k))^(5*k) / (1 - x^k)^k.

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A285247 Expansion of Product_{k>=1} (1-x^(3*k-1))^(3*k-1) * (1-x^(3*k-2))^(3*k-2).

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

A262947 Expansion of Product_{k>=1} 1/(1-x^(3k-2))^(3k-2).

A262924 Expansion of Product_{k>=1} (1 + x^(3k-1))^(3k-1) * (1 + x^(3k-2))^(3k-2).

A262883 Expansion of Product_{k>=1} 1/((1-x^(3k-1))(1-x^(3*k-2)))^k.

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

A285246 Expansion of Product_{k>=1} (1 - x^(5k))^(5k) / (1 - x^k)^k.

A285247 Expansion of Product_{k>=1} (1-x^(3k-1))^(3k-1) * (1-x^(3k-2))^(3k-2).