A261632 -id:A261632 - cp's OEIS frontend

A261629 Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^2.

1, 2, 3, 4, 5, 8, 11, 14, 17, 22, 30, 38, 46, 56, 70, 88, 106, 126, 153, 186, 224, 264, 312, 372, 440, 516, 603, 708, 830, 964, 1117, 1296, 1503, 1734, 1992, 2292, 2638, 3024, 3453, 3942, 4504, 5134, 5831, 6616, 7511, 8518, 9631, 10872, 12274, 13848, 15592

Offset: 0

Vaclav Kotesovec, Aug 27 2015

nonn

Cf. A035451, A261632, A261636.

nmax=60; CoefficientList[Series[Product[1/(1-x^(4*k+1))^2, {k, 0, nmax}], {x, 0, nmax}], x]

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

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)).

A261631 Expansion of Product_{k>=0} 1/(1-x^(3*k+1))^3.

Original entry on oeis.org

1, 3, 6, 10, 18, 30, 46, 69, 105, 154, 219, 309, 434, 597, 813, 1100, 1476, 1959, 2585, 3387, 4410, 5709, 7353, 9414, 12001, 15231, 19242, 24205, 30348, 37902, 47165, 58500, 72342, 89169, 109599, 134337, 164221, 200226, 243537, 295496, 357732, 432117, 520858

Offset: 0

Vaclav Kotesovec, Aug 27 2015

nonn

Cf. A035382, A261616, A261632.

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

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

cp's OEIS Frontend

A261629 Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^2.

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

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A261631 Expansion of Product_{k>=0} 1/(1-x^(3*k+1))^3.

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