A261616 -id:A261616 - cp's OEIS frontend

A261615 Expansion of Product_{k>=0} (1 + x^(3*k+1))^2.

1, 2, 1, 0, 2, 4, 2, 2, 5, 4, 3, 8, 10, 6, 9, 14, 11, 14, 22, 18, 17, 30, 32, 28, 41, 46, 39, 54, 68, 60, 73, 94, 85, 96, 131, 128, 130, 170, 175, 176, 229, 246, 237, 294, 330, 320, 386, 446, 430, 492, 582, 578, 642, 762, 763, 818, 977, 1008, 1061, 1254, 1311

Offset: 0

Vaclav Kotesovec, Aug 26 2015

nonn

Self-convolution of A261612.

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

Cf. A022567, A035382, A261610, A261612, A261616.

nmax = 60; CoefficientList[Series[Product[(1 + x^(3*k+1))^2, {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(2*n)/3) / (2^(11/12) * sqrt(3) * n^(3/4)).

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

Original entry on oeis.org

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)).

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)).

A261635 Expansion of Product_{k>=0} 1/(1-x^(3*k+1))^4.

Original entry on oeis.org

1, 4, 10, 20, 39, 72, 124, 204, 331, 524, 806, 1216, 1813, 2660, 3846, 5500, 7790, 10916, 15158, 20880, 28544, 38736, 52226, 69972, 93200, 123460, 162700, 213340, 278459, 361860, 468252, 603484, 774844, 991220, 1263576, 1605392, 2033172, 2566972, 3231338

Offset: 0

Vaclav Kotesovec, Aug 27 2015

nonn

Cf. A035382, A261616, A261636.

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

a(n) ~ exp(2*Pi*sqrt(2*n)/3) * Gamma(1/3)^4 / (8 * 2^(1/12) * sqrt(3) * Pi^(8/3) * n^(5/12)).

A374018 Expansion of Product_{k>=1} 1 / (1 - x^(3*k-1))^2.

Original entry on oeis.org

1, 0, 2, 0, 3, 2, 4, 4, 7, 6, 13, 10, 19, 18, 27, 30, 42, 44, 63, 66, 91, 100, 130, 144, 187, 206, 263, 294, 364, 412, 506, 568, 696, 782, 943, 1070, 1273, 1444, 1713, 1936, 2285, 2586, 3027, 3428, 3996, 4516, 5243, 5924, 6841, 7730, 8895, 10030, 11512, 12966, 14825, 16696

Offset: 0

Ilya Gutkovskiy, Jun 25 2024

nonn

Cf. A000712, A022567, A035386, A078182, A261616, A374019.

nmax = 55; CoefficientList[Series[Product[1/(1 - x^(3 k - 1))^2, {k, 1, nmax}], {x, 0, nmax}], x]

a(0) = 1; a(n) = (2/n) * Sum_{k=1..n} A078182(k) * a(n-k).
a(n) = Sum_{k=0..n} A035386(k) * A035386(n-k).
a(n) ~ exp(2*Pi*sqrt(n)/3) * Pi^(4/3) / (3^(3/2) * Gamma(1/3)^2 * n^(11/12)). - Vaclav Kotesovec, Jun 25 2024

cp's OEIS Frontend

A261615 Expansion of Product_{k>=0} (1 + x^(3*k+1))^2.

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

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A261635 Expansion of Product_{k>=0} 1/(1-x^(3*k+1))^4.

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A374018 Expansion of Product_{k>=1} 1 / (1 - x^(3*k-1))^2.

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