A262948 -id:A262948 - cp's OEIS frontend

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

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)).

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)).

A285340 Expansion of Product_{k>=0} (1 + x^(5k+4))^(5k+4).

Original entry on oeis.org

1, 0, 0, 0, 4, 0, 0, 0, 6, 9, 0, 0, 4, 36, 14, 0, 1, 54, 92, 19, 0, 36, 228, 202, 24, 9, 272, 702, 358, 29, 158, 1168, 1696, 598, 70, 1027, 3810, 3605, 904, 501, 4600, 10196, 6898, 1408, 3078, 15805, 24104, 12242, 2838, 14103, 46090, 51376, 20566, 9443, 51682

Offset: 0

Seiichi Manyama, Apr 17 2017

nonn

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k-1))^(m*k-1), then a(n, m) ~ exp(2^(-4/3) * 3^(4/3) * m^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(1/6 + 1/(2*m) + m/12) * 3^(1/3) * m^(1/6) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Apr 17 2017

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

Product_{k>=0} (1 + x^(m*k+m-1))^(m*k+m-1): A262736 (m=2), A262948 (m=3), A285339 (m=4), this sequence (m=5).

Cf. A285214, A285338.

nmax = 50; CoefficientList[Series[Product[(1 + x^(5*k-1))^(5*k-1), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 17 2017 *)

a(n) ~ exp(2^(-4/3) * 3^(4/3) * 5^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(41/60) * 3^(1/3) * 5^(1/6) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Apr 17 2017

A285339 Expansion of Product_{k>=0} (1 + x^(4k+3))^(4k+3).

Original entry on oeis.org

1, 0, 0, 3, 0, 0, 3, 7, 0, 1, 21, 11, 0, 21, 54, 15, 7, 96, 122, 19, 74, 311, 217, 44, 351, 768, 367, 209, 1227, 1663, 591, 989, 3402, 3225, 1156, 3609, 8289, 5815, 3053, 11096, 18015, 10176, 9466, 29593, 36249, 18454, 28960, 71093, 68438, 37297, 81606

Offset: 0

Seiichi Manyama, Apr 17 2017

nonn

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

Product_{k>=0} (1 + x^(m*k+m-1))^(m*k+m-1): A262736 (m=2), A262948 (m=3), this sequence (m=4), A285340 (m=5).

Cf. A285213, A285311.

a(n) = (-1)^n * A285213(n).

a(n) ~ exp(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)). - Vaclav Kotesovec, Nov 10 2017

cp's OEIS Frontend

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

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A285340 Expansion of Product_{k>=0} (1 + x^(5*k+4))^(5*k+4).

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A285339 Expansion of Product_{k>=0} (1 + x^(4*k+3))^(4*k+3).

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

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

A285340 Expansion of Product_{k>=0} (1 + x^(5k+4))^(5k+4).

A285339 Expansion of Product_{k>=0} (1 + x^(4k+3))^(4k+3).