A261629 -id:A261629 - cp's OEIS frontend

A261632 Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^3.

1, 3, 6, 10, 15, 24, 37, 54, 75, 103, 144, 198, 265, 348, 456, 599, 777, 993, 1262, 1602, 2028, 2543, 3165, 3930, 4868, 6003, 7359, 8991, 10965, 13329, 16138, 19473, 23448, 28171, 33738, 40293, 48025, 57132, 67803, 80267, 94845, 111888, 131736, 154779, 181530

Offset: 0

Vaclav Kotesovec, Aug 27 2015

nonn

Cf. A035382, A261616, A261631, A035451, A261629.

nmax=50; CoefficientList[Series[Product[1/(1-x^(4*k+1))^3, {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(n/2)) * 2^(3/8) * Gamma(1/4)^3 / (16 * Pi^(9/4) * n^(3/8)).

A261636 Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^4.

Original entry on oeis.org

1, 4, 10, 20, 35, 60, 100, 160, 245, 364, 536, 780, 1115, 1564, 2166, 2980, 4065, 5484, 7326, 9720, 12830, 16824, 21902, 28344, 36510, 46820, 59736, 75844, 95910, 120844, 151688, 189668, 236330, 293564, 363542, 448804, 552425, 678144, 830338, 1014052, 1235296

Offset: 0

Vaclav Kotesovec, Aug 27 2015

nonn

In general, if j > 0, a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} 1/(1 - x^(a*k+b))^j, then a(n) ~ Gamma(b/a)^j * 2^(-(j+5)/4 - j*b/(2*a)) * 3^((j-1)/4 - j*b/(2*a)) * j^(-(j-1)/4 + j*b/(2*a)) * a^(-(j+1)/4 + j*b/(2*a)) * Pi^(-j + j*b/a) * n^((j-3)/4 - j*b/(2*a)) * exp(Pi*sqrt(2*j*n/(3*a))).

Cf. A035382, A261616, A261635, A035451, A261629, A261632.

nmax=50; CoefficientList[Series[Product[1/(1-x^(4*k+1))^4, {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^(1/4) * Gamma(1/4)^4 / (32 * Pi^3 * n^(1/4)).

A374019 Expansion of Product_{k>=1} 1 / (1 - x^(4*k-1))^2.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 3, 2, 0, 4, 4, 2, 5, 6, 7, 8, 8, 12, 15, 12, 17, 26, 23, 24, 37, 40, 39, 50, 62, 66, 74, 86, 101, 116, 122, 144, 175, 184, 202, 246, 274, 294, 340, 388, 432, 480, 533, 610, 684, 742, 835, 956, 1045, 1144, 1299, 1450, 1586, 1758, 1965, 2182, 2400, 2638, 2941, 3268, 3560, 3922

Offset: 0

Ilya Gutkovskiy, Jun 25 2024

nonn

Cf. A000712, A022567, A035462, A050452, A261629, A374018.

nmax = 65; CoefficientList[Series[Product[1/(1 - x^(4 k - 1))^2, {k, 1, nmax}], {x, 0, nmax}], x]

a(0) = 1; a(n) = (2/n) * Sum_{k=1..n} A050452(k) * a(n-k).
a(n) = Sum_{k=0..n} A035462(k) * A035462(n-k).
a(n) ~ Pi^(3/2) * exp(Pi*sqrt(n/3)) / (2*sqrt(3) * Gamma(1/4)^2 * n). - Vaclav Kotesovec, Jun 25 2024

cp's OEIS Frontend

A261632 Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^3.

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A261636 Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^4.

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A374019 Expansion of Product_{k>=1} 1 / (1 - x^(4*k-1))^2.

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