A263138 -id:A263138 - 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)).

A263140 Expansion of Product_{k>=1} (1 + x^(2*k-1))^k.

Original entry on oeis.org

1, 1, 0, 2, 2, 3, 4, 5, 10, 11, 16, 20, 31, 39, 50, 71, 93, 124, 154, 211, 271, 357, 449, 587, 762, 968, 1233, 1571, 2021, 2535, 3220, 4049, 5145, 6431, 8070, 10105, 12670, 15784, 19619, 24447, 30348, 37635, 46464, 57532, 70945, 87477, 107456, 132192, 162220

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, A263149, A263150, A263199, A262878, A263138, A263145, A292037.

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^j/(1 - x^(2*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

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

A263145 Expansion of Product_{k>=1} (1+x^(5*k-1))^k.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, 0, 4, 4, 0, 0, 1, 10, 5, 0, 0, 6, 16, 6, 0, 0, 14, 28, 7, 0, 3, 32, 40, 8, 0, 10, 63, 60, 9, 0, 33, 112, 80, 10, 3, 74, 187, 110, 11, 14, 161, 300, 140, 12, 46, 308, 455, 182, 14, 120, 568, 672, 224, 26, 283

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. A263146, A263147, A263148, A263141, A263140, A262878, A263138.

nmax = 100; CoefficientList[Series[Product[(1+x^(5k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(4*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^(4*j)/(1 - x^(5*j))^2).

a(n) ~ 2^(27/100) * 3^(2/3) * 5^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(32400*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(1/3) * 5^(2/3) * n^(1/3) / (900*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)).

A263136 Expansion of Product_{k>=1} 1/(1-x^(4*k-1))^k.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 2, 0, 1, 2, 3, 1, 2, 6, 5, 2, 6, 11, 7, 6, 15, 21, 12, 15, 30, 34, 22, 35, 58, 59, 43, 70, 108, 95, 85, 142, 187, 161, 167, 263, 318, 274, 318, 480, 534, 471, 595, 836, 879, 819, 1081, 1433, 1442, 1429, 1915, 2391, 2365, 2483, 3314, 3947

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, A262876, A263141, A263137, A263176, A263138.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+4, 4, 'r')=3, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100); # after Alois P. Heinz

nmax = 100; CoefficientList[Series[Product[1/(1-x^(4k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(3*j)/(1 - x^(4*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} 1/j*x^(3*j)/(1 - x^(4*j))^2).

a(n) ~ Zeta(3)^(53/288) * exp(d41 - Pi^4/(6912*Zeta(3)) + Pi^2 * n^(1/3) / (48*Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3)/4) / (sqrt(3*Pi) * 2^(101/96) * n^(197/288)), where d41 = A263176 = Integral_{x=0..infinity} exp(-3*x)/(x*(1 - exp(-4*x))^2) - 1/(16*x^3) - 1/(16*x^2) + 5/(96*x*exp(x)) = -0.158924147180165035059952001737321408554746599955833696821824808027... .

A263139 Expansion of Product_{k>=1} (1+x^(4*k-3))^k.

Original entry on oeis.org

1, 1, 0, 0, 0, 2, 2, 0, 0, 3, 4, 1, 0, 4, 10, 6, 0, 5, 16, 14, 3, 6, 28, 32, 10, 7, 40, 63, 33, 11, 60, 112, 74, 23, 80, 187, 161, 56, 111, 300, 308, 131, 152, 455, 568, 295, 223, 672, 968, 607, 356, 967, 1609, 1186, 618, 1367, 2546, 2189, 1132, 1926, 3941

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. A263138, A263137, A263147.

nmax = 100; CoefficientList[Series[Product[(1+x^(4k-3))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^j/(1 - x^(4*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^j/(1 - x^(4*j))^2).

a(n) ~ 2^(83/96) * 3^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(2304*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(2/3) * n^(1/3) / (96*Zeta(3)^(1/3)) + Zeta(3)^(1/3) * 2^(-8/3) * 3^(4/3) * n^(2/3)) / (12 * sqrt(Pi) * n^(2/3)).

cp's OEIS Frontend

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

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A263140 Expansion of Product_{k>=1} (1 + x^(2*k-1))^k.

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A263145 Expansion of Product_{k>=1} (1+x^(5*k-1))^k.

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A263136 Expansion of Product_{k>=1} 1/(1-x^(4*k-1))^k.

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A263139 Expansion of Product_{k>=1} (1+x^(4*k-3))^k.

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