A262883 -id:A262883 - cp's OEIS frontend

A262876 Expansion of Product_{k>=1} 1/(1-x^(3*k-1))^k.

1, 0, 1, 0, 1, 2, 1, 2, 4, 2, 7, 6, 7, 12, 12, 16, 26, 22, 35, 44, 47, 68, 84, 88, 133, 146, 176, 238, 267, 324, 431, 468, 604, 746, 842, 1068, 1296, 1470, 1884, 2202, 2579, 3220, 3753, 4418, 5483, 6294, 7541, 9144, 10554, 12644, 15191, 17480, 21057, 24896

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

a(n) is the number of partitions of n into parts 3*k-1 of k kinds (k>=1).

In general, if s>0, t>0, GCD(s,t)=1 and g.f. = Product_{k>=1} 1/(1 - x^(s*k-t))^k then a(n) ~ s^(t^2/(3*s^2) - 7/18) * n^(t^2/(6*s^2) - 25/36) * exp(d(s,t) - Pi^4 * t^2 / (432*s^2 * Zeta(3)) + Pi^2 * t * 2^(2/3) * s^(2/3) * n^(1/3) / (12 * s^2 * Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / (2^(2/3)*s^(2/3))) / (2^(t^2/(6*s^2) + 11/36) * sqrt(3*Pi) * Zeta(3)^(t^2/(6*s^2) - 7/36)), where d(s,t) = Integral_{x=0..infinity} 1/x * (exp(-(s-t)*x)/(1 - exp(-s*x))^2 - 1/(s^2*x^2) - t/(s^2*x) + exp(-x)*(1/12 - t^2/(2*s^2))) dx. - Vaclav Kotesovec, Oct 12 2015

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
Vaclav Kotesovec, Graph - The asymptotic ratio (70000 terms)

Cf. A000219, A035528, A255610, A262877, A262878, A262879, A262883, A262946, A263136, A263141.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+3, 3, 'r')=2, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 05 2015

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

a(n) ~ Zeta(3)^(19/108) * exp(d1 - Pi^4 / (3888*Zeta(3)) + Pi^2 * n^(1/3) / (2^(4/3)*3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (2^(35/108) * 3^(23/27) * sqrt(Pi) * n^(73/108)), 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) = -0.188708191979528532376410098649207973592114467268429221509... . - Vaclav Kotesovec, Oct 08 2015

A262877 Expansion of Product_{k>=1} 1/(1-x^(3*k-2))^k.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 6, 9, 9, 13, 19, 23, 28, 42, 51, 62, 84, 108, 127, 170, 219, 261, 328, 427, 512, 632, 807, 987, 1190, 1504, 1838, 2214, 2744, 3374, 4036, 4950, 6060, 7260, 8793, 10748, 12853, 15459, 18766, 22473, 26834, 32425, 38768, 46136, 55376, 66168

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

a(n) is the number of partitions of n into parts 3*k-2 of k kinds (k>=1). - Joerg Arndt, Oct 06 2015

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
Vaclav Kotesovec, Graph - The asymptotic ratio (70000 terms)

Cf. A000219, A035528, A255610, A262876, A262878, A262879, A262883, A262947.

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

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

a(n) ~ Zeta(3)^(13/108) * exp(d2 - Pi^4 / (972*Zeta(3)) + Pi^2 * n^(1/3) / (2^(1/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (2^(41/108) * 3^(20/27) * sqrt(Pi) * n^(67/108)), 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) = -0.0145374291832840336050202945022620903605414975934644413815... . - Vaclav Kotesovec, Oct 08 2015

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

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

A262923 Expansion of Product_{k>=1} 1 / ((1-x^(3k-1))^(3k-1) * (1-x^(3k-2))^(3k-2)).

Original entry on oeis.org

1, 1, 3, 3, 10, 15, 27, 44, 79, 128, 211, 331, 549, 843, 1338, 2061, 3195, 4851, 7384, 11104, 16696, 24774, 36817, 54173, 79560, 116067, 168880, 244293, 352480, 506012, 724531, 1032762, 1468271, 2079525, 2937102, 4134399, 5804795, 8124459, 11342952, 15791650

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

Convolution of A262946 and A262947.

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. A262883, A262876, A262877, A262924, A035386, A035382, A262946, A262947.

nmax=60; CoefficientList[Series[Product[1/((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(-1/6 + 3^(2/3)*(Zeta(3)/2)^(1/3) * n^(2/3)) * A^2 * Zeta(3)^(1/9) / (2^(5/18) * 3^(31/36) * sqrt(Pi) * n^(11/18)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A262884 Expansion of Product_{k>=1} ((1+x^(3k-1))(1+x^(3*k-2)))^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 4, 7, 9, 11, 16, 23, 31, 40, 53, 71, 91, 121, 161, 206, 264, 343, 441, 563, 725, 922, 1166, 1476, 1869, 2357, 2967, 3725, 4659, 5816, 7263, 9050, 11241, 13947, 17269, 21333, 26342, 32479, 39957, 49094, 60231, 73775, 90273, 110333, 134643

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

Convolution of A262878 and A262879.

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
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. A262878, A262879, A262883, A262924.

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

a(n) ~ exp(-Pi^4/(2592*Zeta(3)) + Pi^2 * n^(1/3) / (12*3^(2/3)*Zeta(3)^(1/3)) + 3^(2/3) * Zeta(3)^(1/3) * n^(2/3)/2) * Zeta(3)^(1/6) / (2^(7/18) * 3^(2/3) * sqrt(Pi) * n^(2/3)).

cp's OEIS Frontend

A262876 Expansion of Product_{k>=1} 1/(1-x^(3*k-1))^k.

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A262877 Expansion of Product_{k>=1} 1/(1-x^(3*k-2))^k.

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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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A262923 Expansion of Product_{k>=1} 1 / ((1-x^(3*k-1))^(3*k-1) * (1-x^(3*k-2))^(3*k-2)).

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A262884 Expansion of Product_{k>=1} ((1+x^(3*k-1))*(1+x^(3*k-2)))^k.

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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).

A262923 Expansion of Product_{k>=1} 1 / ((1-x^(3k-1))^(3k-1) * (1-x^(3k-2))^(3k-2)).

A262884 Expansion of Product_{k>=1} ((1+x^(3k-1))(1+x^(3*k-2)))^k.