A285214 -id:A285214 - cp's OEIS frontend

A285213 Expansion of Product_{k>=0} (1-x^(4k+3))^(4k+3).

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

Offset: 0

Seiichi Manyama, Apr 15 2017

sign

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

Product_{k>=0} (1-x^(m*k+m-1))^(m*k+m-1): A285069 (m=2), A285212 (m=3), this sequence (m=4), A285214 (m=5).

Cf. A285131.

nmax = 50; CoefficientList[Series[Product[(1-x^(4*k-1))^(4*k-1), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

x='x+O('x^100); Vec(prod(k=0, 100, (1 - x^(4*k + 3))^(4*k + 3))) \\ Indranil Ghosh, Apr 15 2017

a(n) ~ (-1)^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, Apr 17 2017

A285212 Expansion of Product_{k>=0} (1-x^(3k+2))^(3k+2).

Original entry on oeis.org

1, 0, -2, 0, 1, -5, 0, 10, -8, -5, 26, -11, -28, 62, -4, -101, 111, 43, -260, 182, 228, -583, 202, 715, -1155, 25, 1888, -2034, -851, 4286, -3144, -3418, 8895, -3888, -9806, 16848, -2479, -23812, 29519, 5626, -52156, 46930, 30033, -105320, 66001, 90431, -198736

Offset: 0

Seiichi Manyama, Apr 15 2017

sign

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

Product_{k>=0} (1-x^(m*k+m-1))^(m*k+m-1): A285069 (m=2), this sequence (m=3), A285213 (m=4), A285214 (m=5).

Cf. A262946.

nmax = 100; CoefficientList[Series[Product[(1-x^(3*k+2))^(3*k+2), {k,0,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

x='x+O('x^100); Vec(prod(k=0, 100, (1 - x^(3*k + 2))^(3*k + 2))) \\ Indranil Ghosh, Apr 15 2017

A285132 Expansion of Product_{k>=0} 1/(1-x^(5k+4))^(5k+4).

Original entry on oeis.org

1, 0, 0, 0, 4, 0, 0, 0, 10, 9, 0, 0, 20, 36, 14, 0, 35, 90, 101, 19, 56, 180, 320, 202, 108, 315, 730, 859, 492, 533, 1390, 2300, 2139, 1354, 2393, 4835, 6475, 5098, 4619, 8813, 14926, 16395, 12982, 15751, 28962, 41162, 40256, 35200, 51731, 85365, 106145

Offset: 0

Seiichi Manyama, Apr 15 2017

nonn

In general, if m > 1 and g.f. = Product_{k>=1} 1/(1-x^(m*k-1))^(m*k-1), then a(n, m) ~ exp(c*m + 3 * 2^(-2/3) * m^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * (2*Zeta(3))^(1/(6*m) + m/36) / (sqrt(3) * Gamma(1 - 1/m) * m^(1/2 - 5/(6*m) + m/36) * n^(1/2 + 1/(6*m) + m/36)), where c = Integral_{x=0..infinity} exp((m+1)*x) / (x*(exp(m*x)-1)^2) + (1/12 - 1/(2*m^2))/(x*exp(x)) - 1/(m^2*x^3) - 1/(m^2*x^2) dx. - Vaclav Kotesovec, Apr 17 2017

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

Product_{k>=0} 1/(1-x^(m*k+m-1))^(m*k+m-1): A262811 (m=2), A262946 (m=3), A285131 (m=4), this sequence (m=5).

Cf. A285214.

nmax = 100; CoefficientList[Series[Product[1/(1-x^(5*k-1))^(5*k-1), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

x='x+O('x^100); Vec(prod(k=0, 100, 1/(1 - x^(5*k + 4))^(5*k + 4))) \\ Indranil Ghosh, Apr 15 2017

a(n) ~ exp(5*c + 3*2^(-2/3)*5^(-1/3)*Zeta(3)^(1/3)*n^(2/3)) * (2*Zeta(3))^(31/180) / (sqrt(3) * 5^(17/36) * Gamma(4/5) * n^(121/180)), where c = Integral_{x=0..inf} ((19/(exp(x)*300) + 1/(exp(4*x)*(1-exp(-5*x))^2) - 1/(25*x^2) - 1/(25*x))/x) dx = -0.12699586713882325294527057473113580561183418857868946729897216431919... - Vaclav Kotesovec, Apr 15 2017

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

cp's OEIS Frontend

A285213 Expansion of Product_{k>=0} (1-x^(4*k+3))^(4*k+3).

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A285212 Expansion of Product_{k>=0} (1-x^(3*k+2))^(3*k+2).

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A285132 Expansion of Product_{k>=0} 1/(1-x^(5*k+4))^(5*k+4).

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

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A285213 Expansion of Product_{k>=0} (1-x^(4k+3))^(4k+3).

A285212 Expansion of Product_{k>=0} (1-x^(3k+2))^(3k+2).

A285132 Expansion of Product_{k>=0} 1/(1-x^(5k+4))^(5k+4).

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