A280277 -id:A280277 - cp's OEIS frontend

A280276 G.f.: Product_{k>=1} (1 + x^k) / (1 - x^(k^2)).

1, 2, 3, 5, 8, 12, 17, 24, 33, 46, 62, 82, 108, 141, 182, 233, 297, 375, 472, 590, 733, 907, 1117, 1369, 1671, 2034, 2465, 2978, 3586, 4304, 5152, 6149, 7319, 8689, 10293, 12162, 14340, 16871, 19806, 23207, 27139, 31678, 36909, 42932, 49851, 57794, 66897

Offset: 0

Vaclav Kotesovec, Dec 30 2016

nonn

Convolution of A000009 and A001156.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A001156, A087154, A103265, A280277, A369570, A369574.

nmax=80; CoefficientList[Series[Product[(1+x^k)/(1-x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(n/3) + 3^(1/4) * Zeta(3/2) * n^(1/4) / sqrt(2) - 3*Zeta(3/2)^2 / (16*Pi)) / (8*sqrt(6*Pi)*n).

A369571 Expansion of Product_{k>=1} (1 + x^(k^3)) * (1 + x^k).

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 20, 25, 31, 38, 47, 58, 70, 84, 102, 122, 145, 173, 205, 242, 285, 334, 391, 458, 534, 620, 720, 833, 961, 1109, 1276, 1466, 1683, 1926, 2201, 2513, 2863, 3258, 3704, 4203, 4763, 5394, 6098, 6885, 7768, 8752, 9850, 11076, 12439

Offset: 0

Vaclav Kotesovec, Jan 26 2024

nonn

Convolution of A279329 and A000009.

a(n) is the number of pairs (Q(k), P(n-k)), 0<=k<=n, where Q(k) is a partition of k into distinct cubes and P(n-k) is a partition of n-k into distinct parts.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000009, A279329, A280278, A280277.

nmax = 100; CoefficientList[Series[Product[(1+x^(k^3))*(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(n/3) + (2^(1/3) - 1) * Gamma(1/3) * zeta(4/3) * n^(1/6) / (3^(5/6) * Pi^(1/3))) / (2^(5/2) * 3^(1/4) * n^(3/4)).

A369573 Expansion of Product_{k>=1} (1 + x^(k^2)) / (1 - x^(k^3)).

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 4, 4, 5, 7, 8, 8, 9, 11, 12, 12, 14, 17, 18, 18, 20, 23, 24, 24, 26, 31, 34, 35, 38, 43, 46, 47, 50, 55, 59, 61, 66, 73, 77, 79, 84, 92, 97, 100, 106, 115, 121, 124, 130, 140, 148, 152, 161, 174, 183, 188, 197, 210, 220, 226, 235, 251, 264, 272

Offset: 0

Vaclav Kotesovec, Jan 26 2024

nonn

Convolution of A033461 and A003108.

a(n) is the number of pairs (Q(k), P(n-k)), 0<=k<=n, where Q(k) is a partition of k into distinct squares and P(n-k) is a partition of n-k into cubes.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A003108, A033461, A280204, A280277.

nmax = 100; CoefficientList[Series[Product[(1+x^(k^2))/(1-x^(k^3)), {k, 1, nmax^(1/2)}], {x, 0, nmax}], x]

cp's OEIS Frontend

A280276 G.f.: Product_{k>=1} (1 + x^k) / (1 - x^(k^2)).

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A369571 Expansion of Product_{k>=1} (1 + x^(k^3)) * (1 + x^k).

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Formula

A369573 Expansion of Product_{k>=1} (1 + x^(k^2)) / (1 - x^(k^3)).

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Author

Keywords

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Mathematica