A263145 -id:A263145 - 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)).

A263141 Expansion of Product_{k>=1} 1/(1-x^(5*k-1))^k.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 2, 3, 0, 1, 2, 6, 4, 1, 2, 6, 10, 6, 2, 6, 14, 20, 8, 6, 14, 29, 30, 13, 14, 34, 54, 50, 22, 34, 66, 99, 74, 43, 72, 133, 166, 119, 82, 148, 242, 276, 182, 166, 286, 438, 442, 301, 316, 541, 744, 701, 494, 608, 976, 1255

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. A263142, A263143, A263144, A263178, A263145, A035528, A262876, A263136.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+5, 5, 'r')=4, 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^(5k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[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*x^(4*j)/(1 - x^(5*j))^2).

a(n) ~ Zeta(3)^(169/900) * exp(d51 - Pi^4/(10800*Zeta(3))+ Pi^2 * 2^(2/3) * 5^(2/3) * n^(1/3) / (300 * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * 2^(-2/3) * 5^(-2/3) * n^(2/3)) / (2^(281/900) * 5^(169/450) * sqrt(3*Pi) * n^(619/900)), where d51 = A263178 = Integral_{x=0..infinity} exp(-4*x)/(x*(1 - exp(-5*x))^2) - 1/(25*x^3) - 1/(25*x^2) + 19/(300*x*exp(x)) = -0.1269958671388232529452705747311358056... .

A263138 Expansion of Product_{k>=1} (1+x^(4*k-1))^k.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 3, 0, 0, 4, 4, 0, 1, 10, 5, 0, 6, 16, 6, 0, 14, 28, 7, 3, 32, 40, 8, 10, 63, 60, 9, 33, 112, 80, 13, 74, 187, 110, 25, 161, 300, 140, 58, 308, 455, 183, 133, 568, 672, 236, 297, 968, 963, 321, 609, 1609, 1344, 468, 1188, 2546

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. A263139, A263136, A263140, A262878, A263145.

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

a(n) ~ 2^(59/96) * 3^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(20736*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(2/3) * n^(1/3) / (288*Zeta(3)^(1/3)) + Zeta(3)^(1/3) * 2^(-8/3) * 3^(4/3) * n^(2/3)) / (12 * sqrt(Pi) * n^(2/3)).

A263147 Expansion of Product_{k>=1} (1+x^(5*k-3))^k.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 0, 3, 0, 4, 0, 1, 4, 0, 10, 0, 6, 5, 0, 16, 0, 14, 6, 3, 28, 0, 32, 7, 10, 40, 0, 63, 8, 33, 60, 3, 112, 9, 74, 80, 14, 187, 10, 161, 110, 46, 300, 12, 308, 140, 120, 455, 24, 568, 182, 283, 672, 54, 968, 224, 594, 963, 146

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, A263146, A263148, A263143.

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

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

a(n) ~ 2^(43/100) * 3^(2/3) * 5^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(3600*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(1/3) * 5^(2/3) * n^(1/3) / (300*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)).

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)).

A263148 Expansion of Product_{k>=1} (1+x^(5*k-4))^k.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 3, 4, 1, 0, 0, 4, 10, 6, 0, 0, 5, 16, 14, 3, 0, 6, 28, 32, 10, 0, 7, 40, 63, 33, 3, 8, 60, 112, 74, 14, 9, 80, 187, 161, 46, 11, 110, 300, 308, 120, 23, 140, 455, 568, 283, 53, 182, 672, 968, 594, 145, 228, 963, 1609, 1172

Offset: 0

Vaclav Kotesovec, Oct 10 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, 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, A263146, A263147, A263144.

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

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

a(n) ~ 2^(57/100) * 3^(2/3) * 5^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(2025*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(1/3) * 5^(2/3) * n^(1/3) / (225*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)).

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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A263141 Expansion of Product_{k>=1} 1/(1-x^(5*k-1))^k.

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A263138 Expansion of Product_{k>=1} (1+x^(4*k-1))^k.

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A263147 Expansion of Product_{k>=1} (1+x^(5*k-3))^k.

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

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A263148 Expansion of Product_{k>=1} (1+x^(5*k-4))^k.

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