A280276 -id:A280276 - cp's OEIS frontend

A369570 Expansion of Product_{k>=1} (1 + x^(k^2)) * (1 + x^k).

1, 2, 2, 3, 5, 7, 9, 12, 15, 20, 27, 33, 41, 52, 65, 80, 99, 120, 145, 177, 213, 255, 305, 363, 430, 511, 604, 709, 833, 976, 1141, 1331, 1547, 1793, 2079, 2406, 2775, 3197, 3676, 4221, 4841, 5541, 6330, 7225, 8235, 9372, 10655, 12094, 13710, 15529, 17568, 19848

Offset: 0

Vaclav Kotesovec, Jan 26 2024

nonn

Convolution of A033461 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 squares 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, A033461, A280204, A280276.

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

a(n) ~ exp(Pi*sqrt(n/3) + 3^(1/4) * (sqrt(2) - 1) * zeta(3/2) * n^(1/4)/2 - 3*(3 - 2*sqrt(2)) * zeta(3/2)^2/(32*Pi)) / (2^(5/2) * 3^(1/4) * n^(3/4)).

A280277 G.f.: Product_{k>=1} (1 + x^k) / (1 - x^(k^3)).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 14, 19, 26, 35, 46, 60, 77, 98, 124, 156, 195, 242, 299, 367, 448, 545, 660, 796, 957, 1146, 1368, 1629, 1933, 2287, 2700, 3178, 3732, 4373, 5112, 5964, 6944, 8068, 9357, 10832, 12517, 14440, 16632, 19126, 21960, 25178, 28825, 32954, 37625

Offset: 0

Vaclav Kotesovec, Dec 30 2016

nonn

Convolution of A000009 and A003108.

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

Cf. A000009, A003108, A102108, A280264, A280276, A369571, A369573.

nmax=80; CoefficientList[Series[Product[(1+x^k)/(1-x^(k^3)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(n/3) + 2^(1/3) * Gamma(1/3) * Zeta(4/3) * n^(1/6) / (3^(5/6) * Pi^(1/3))) / (16*sqrt(3)*Pi*n).

A280422 G.f.: Product_{k>=1} (1 + x^k) / (1 - x^(k*(k+1)/2)).

Original entry on oeis.org

1, 2, 3, 6, 9, 13, 21, 30, 41, 59, 81, 108, 147, 195, 253, 333, 431, 549, 704, 892, 1119, 1409, 1758, 2176, 2697, 3321, 4065, 4976, 6061, 7345, 8898, 10737, 12901, 15489, 18535, 22103, 26333, 31284, 37056, 43844, 51751, 60931, 71655, 84090, 98464, 115162

Offset: 0

Vaclav Kotesovec, Jan 02 2017

nonn

Convolution of A007294 and A000009.

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

Cf. A007294, A000009, A280276, A280424.

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

a(n) ~ exp(sqrt(n/3)*Pi + 3^(1/4) * Zeta(3/2) * n^(1/4) - 3*Zeta(3/2)^2/(8*Pi)) / (32 * 3^(3/4) * n^(5/4)).

A369574 Expansion of Product_{k>=1} (1 + x^(k^3)) / (1 - x^(k^2)).

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 4, 4, 6, 9, 10, 10, 12, 15, 16, 16, 19, 24, 27, 28, 31, 36, 39, 40, 44, 52, 58, 62, 68, 76, 82, 86, 93, 104, 114, 122, 134, 148, 158, 166, 179, 196, 210, 223, 242, 265, 282, 295, 315, 342, 365, 384, 412, 447, 476, 498, 527, 566, 602, 632, 670

Offset: 0

Vaclav Kotesovec, Jan 26 2024

nonn

Convolution of A279329 and A001156.

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 squares.

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

Cf. A001156, A279329, A280278, A280276.

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

a(n) ~ exp((27*zeta(3/2)^(8/9) * (729*(48 + 6*2^(1/3) - 35*2^(2/3)) * Pi^(4/3) * zeta(3/2)^(16/9) * n^(1/3) + 972 * 2^(1/9)*(41 - 59*2^(1/3) + 21*2^(2/3)) * Gamma(4/3) * zeta(4/3) * (Pi*zeta(3/2))^(8/9) * n^(2/9) - 8*2^(2/9)*(-160 + 2^(1/3) + 100*2^(2/3)) * Pi^(4/9) * Gamma(1/3)^2 * zeta(4/3)^2 * n^(1/9)) + 32*(-161 - 99*2^(1/3) + 180*2^(2/3)) * Gamma(1/3)^3 * zeta(4/3)^3) / (26244*(-1 + 2^(1/3))^6 * Pi * zeta(3/2)^2)) * zeta(3/2)^(2/3) * (-198 + 360*2^(1/3) - 161*2^(2/3)) / (8*sqrt(6) * (-1 + 2^(1/3))^9 * Pi^(7/6) * n^(7/6)).

cp's OEIS Frontend

A369570 Expansion of Product_{k>=1} (1 + x^(k^2)) * (1 + x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A280277 G.f.: Product_{k>=1} (1 + x^k) / (1 - x^(k^3)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A280422 G.f.: Product_{k>=1} (1 + x^k) / (1 - x^(k*(k+1)/2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A369574 Expansion of Product_{k>=1} (1 + x^(k^3)) / (1 - x^(k^2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula