A261612 -id:A261612 - cp's OEIS frontend

A262953 Expansion of Product_{k>=1} (1 + x^(2k-1)) (1 + x^(3*k-2)).

1, 2, 1, 1, 3, 4, 3, 4, 7, 7, 8, 11, 13, 15, 18, 21, 25, 31, 34, 38, 48, 56, 61, 72, 85, 95, 109, 126, 142, 163, 186, 207, 237, 272, 301, 339, 389, 433, 482, 547, 612, 680, 764, 851, 946, 1061, 1177, 1301, 1455, 1616, 1779, 1977, 2194, 2415, 2670, 2953, 3250

Offset: 0

Vaclav Kotesovec, Oct 05 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
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. A000700, A261612, A262952.

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

a(n) ~ 5^(1/4) * exp(Pi*sqrt(5*n/2)/3) / (2^(19/12) * sqrt(3) * n^(3/4)).

A284095 Expansion of Product_{k>=0} (1 + x^(8*k+1)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 0, 1, 4, 5, 2, 0, 0, 0, 0, 1, 5, 7, 3, 0, 0, 0, 0, 1, 5, 8, 5, 1, 0, 0, 0

Offset: 0

Seiichi Manyama, Mar 20 2017

nonn

Number of partitions of n into distinct parts congruent to 1 mod 8.

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000

Cf. Product_{k>=0} (1 + x^(m*k+1)): A261612 (m=3), A169975 (m=4), A280454 (m=5), A280456 (m=6), A280457 (m=7), this sequence (m=8).

CoefficientList[Series[Product[(1 + x^(8*k + 1)) , {k, 0, 91}], {x, 0, 91}], x] (* Indranil Ghosh, Mar 20 2017 *)
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 8] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Mar 20 2017 *)

Vec(prod(k=0, 91, (1 + x^(8*k + 1))) + O(x^92)) \\ Indranil Ghosh, Mar 20 2017

a(n) ~ exp(sqrt(n/6)*Pi/2) / (2^(15/8) * 3^(1/4) * n^(3/4)) * (1 + (11*Pi/(384*sqrt(6)) - 3*sqrt(3/2)/(2*Pi))/sqrt(n)). - Vaclav Kotesovec, Mar 20 2017

G.f.: Sum_{k>=0} x^(k*(4*k - 3)) / Product_{j=1..k} (1 - x^(8*j)). - Ilya Gutkovskiy, Nov 24 2020

A261633 Expansion of Product_{k>=0} (1+x^(3*k+1))^3.

Original entry on oeis.org

1, 3, 3, 1, 3, 9, 9, 6, 12, 18, 15, 21, 37, 36, 33, 55, 69, 66, 90, 117, 111, 136, 195, 204, 217, 294, 330, 342, 447, 531, 541, 657, 804, 839, 978, 1215, 1288, 1419, 1755, 1936, 2094, 2535, 2846, 3027, 3588, 4123, 4392, 5073, 5869, 6261, 7080

Offset: 0

Vaclav Kotesovec, Aug 27 2015

nonn

Cf. A261612, A261615, A261634.

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

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

A261637 Expansion of Product_{k>=0} (1+x^(3*k+1))^4.

Original entry on oeis.org

1, 4, 6, 4, 5, 16, 24, 20, 26, 48, 56, 60, 98, 132, 130, 168, 253, 288, 324, 448, 539, 588, 780, 1000, 1086, 1304, 1690, 1916, 2205, 2812, 3274, 3648, 4507, 5396, 6018, 7200, 8674, 9680, 11250, 13616, 15425, 17532, 20962, 23988, 26960, 31868, 36836, 41164

Offset: 0

Vaclav Kotesovec, Aug 27 2015

nonn

Cf. A261612, A261615, A261638.

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

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

cp's OEIS Frontend

A262953 Expansion of Product_{k>=1} (1 + x^(2k-1)) (1 + x^(3*k-2)).

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A284095 Expansion of Product_{k>=0} (1 + x^(8*k+1)).

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A261633 Expansion of Product_{k>=0} (1+x^(3*k+1))^3.

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A261637 Expansion of Product_{k>=0} (1+x^(3*k+1))^4.

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cp's OEIS Frontend

A262953 Expansion of Product_{k>=1} (1 + x^(2*k-1)) * (1 + x^(3*k-2)).

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A284095 Expansion of Product_{k>=0} (1 + x^(8*k+1)).

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A261633 Expansion of Product_{k>=0} (1+x^(3*k+1))^3.

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A261637 Expansion of Product_{k>=0} (1+x^(3*k+1))^4.

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A262953 Expansion of Product_{k>=1} (1 + x^(2k-1)) (1 + x^(3*k-2)).