A262879 -id:A262879 - 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

A262878 Expansion of Product_{k>=1} (1+x^(3*k-1))^k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 2, 3, 0, 4, 4, 1, 10, 5, 6, 16, 6, 14, 28, 10, 32, 40, 18, 63, 60, 42, 112, 83, 84, 187, 124, 172, 300, 186, 320, 456, 302, 581, 684, 507, 982, 1004, 874, 1624, 1476, 1508, 2566, 2174, 2582, 3981, 3262, 4338, 6002, 4945, 7138, 8947, 7660

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

In general, if s>0, t>0, GCD(s,t)=1 and g.f. = Product_{k>=1} (1 + x^(s*k-t))^k then a(n) ~ 2^(t^2/(2*s^2) - 3/4) * s^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4 * t^2 / (1296 * s^2 * Zeta(3)) + Pi^2 * t * 2^(1/3) * 3^(2/3) * s^(2/3) * n^(1/3) / (36 * s^2 * Zeta(3)^(1/3)) + 3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / (2^(4/3) * s^(2/3)) ) / (3^(1/3) * s * sqrt(Pi) * n^(2/3)). - 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

Cf. A026007, A027346, A035528, A262876, A262877, A262879, A262884, A262948, A263138, A263145.

with(numtheory):
b:= n-> `if`(n<3, n-1, (p-> [0, -r, 2*r, 0, 0, 2*r+1][p]
         )(1+irem(n+3, 6, 'r'))):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*b(d), 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+x^(3k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax=100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(2*j)/(1-x^(3j))^2,{j,1,nmax}],{x,0,nmax}],x]

a(n) ~ exp(2^(-4/3) * 3^(2/3) * Zeta(3)^(1/3) * n^(2/3) + Pi^2 * n^(1/3) / (2^(5/3)*3^(8/3) * Zeta(3)^(1/3)) - Pi^4/(11664*Zeta(3))) * Zeta(3)^(1/6) / (2^(25/36) * 3^(2/3) * sqrt(Pi) * n^(2/3)).

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)).

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)).

A262949 Expansion of Product_{k>=1} (1 + x^(3k-2))^(3k-2).

Original entry on oeis.org

1, 1, 0, 0, 4, 4, 0, 7, 13, 6, 10, 38, 32, 17, 74, 103, 59, 139, 266, 191, 247, 593, 581, 513, 1175, 1487, 1190, 2223, 3453, 2938, 4158, 7264, 7095, 8052, 14430, 16308, 16246, 27364, 35347, 34096, 50997, 72595, 72163, 94707, 142522, 151435, 178047, 270112

Offset: 0

Vaclav Kotesovec, Oct 05 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A261612, A262879, A262884, A262924, A262948.

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

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

A263146 Expansion of Product_{k>=1} (1+x^(5*k-2))^k.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 0, 3, 0, 0, 4, 0, 4, 1, 0, 10, 0, 5, 6, 0, 16, 0, 6, 14, 0, 28, 3, 7, 32, 0, 40, 10, 8, 63, 0, 60, 33, 9, 112, 3, 80, 74, 10, 187, 14, 110, 161, 11, 300, 46, 140, 308, 13, 455, 120, 182, 568, 25, 672, 283, 224, 968, 55, 963

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. A263145, A263147, A263148, A263142, A262879.

nmax = 100; CoefficientList[Series[Product[(1+x^(5k-2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(3*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^(3*j)/(1 - x^(5*j))^2).

a(n) ~ 2^(33/100) * 3^(2/3) * 5^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(8100*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(1/3) * 5^(2/3) * n^(1/3) / (450*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)).

A263345 Expansion of Product_{k>=1} ((1 + x^k)/(1 + x^(3*k)))^k.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 22, 40, 65, 107, 176, 282, 448, 705, 1101, 1701, 2611, 3977, 6021, 9048, 13527, 20102, 29720, 43712, 63997, 93259, 135317, 195539, 281440, 403559, 576568, 820888, 1164826, 1647583, 2323169, 3266041, 4578305, 6399990, 8922389, 12406535

Offset: 0

Vaclav Kotesovec, Oct 15 2015

nonn

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

Cf. A003105, A026007, A262876, A262877, A262878, A262879, A263346.

Cf. A285289, A285290, A285291.

nmax=40; CoefficientList[Series[Product[((1 + x^k)/(1 + x^(3*k)))^k,{k,1,nmax}],{x,0,nmax}],x]

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

A263346 Expansion of Product_{k>=1} ((1 - x^(3*k))/(1 - x^k))^k.

Original entry on oeis.org

1, 1, 3, 5, 12, 21, 40, 71, 130, 221, 387, 648, 1095, 1800, 2964, 4792, 7730, 12301, 19510, 30619, 47859, 74179, 114469, 175427, 267684, 406039, 613325, 921671, 1379500, 2055313, 3050652, 4509385, 6641966, 9746452, 14254242, 20775255, 30184451, 43715711

Offset: 0

Vaclav Kotesovec, Oct 15 2015

nonn

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

Cf. A000726, A000219, A262876, A262877, A262878, A262879, A263345.

nmax=40; CoefficientList[Series[Product[((1 - x^(3*k))/(1 - x^k))^k,{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ 2^(1/6) * Zeta(3)^(1/6) * exp(6^(1/3) * Zeta(3)^(1/3) * n^(2/3)) / (3^(11/12) * 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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A262878 Expansion of Product_{k>=1} (1+x^(3*k-1))^k.

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

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A262949 Expansion of Product_{k>=1} (1 + x^(3*k-2))^(3*k-2).

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A263146 Expansion of Product_{k>=1} (1+x^(5*k-2))^k.

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A263345 Expansion of Product_{k>=1} ((1 + x^k)/(1 + x^(3*k)))^k.

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A262924 Expansion of Product_{k>=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.

A262949 Expansion of Product_{k>=1} (1 + x^(3k-2))^(3k-2).