A305102 -id:A305102 - cp's OEIS frontend

A305122 G.f.: Sum_{k>=1} x^(2k)/(1+x^(2k)) * Product_{k>=1} (1+x^k)/(1-x^k).

0, 0, 1, 2, 4, 8, 16, 28, 47, 78, 126, 198, 306, 464, 694, 1024, 1490, 2146, 3061, 4322, 6052, 8408, 11592, 15872, 21592, 29192, 39242, 52468, 69788, 92376, 121716, 159664, 208569, 271372, 351732, 454228, 584546, 749720, 958472, 1221560, 1552210, 1966698

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

Convolution of A305121 and A000009.

The g.f. Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) * Product_{k >= 1} (1 + x ^k)/(1 - x^k) = Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) * Product_{k >= 1} (1 + x ^k)/(1 + x^k - 2*x^k) is congruent mod 2 to Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) = -G(-x^2), where G(x) is the g.f. of A112329. It follows that a(n) is odd iff n = 2*k^2 for some positive integer k. - Peter Bala, Jan 07 2025

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

Cf. A305102, A305101, A305104, A305105, A305124.

Cf. A116680, A112329, A305121.

nmax = 50; CoefficientList[Series[Sum[x^(2*k)/(1+x^(2*k)), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) = A305101(n) - A305124(n).

a(n) ~ exp(sqrt(n)*Pi) * log(2) / (8*Pi*sqrt(n)).

A305124 G.f.: Sum_{k>=1} x^(2k-1)/(1+x^(2k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

Original entry on oeis.org

0, 1, 1, 4, 7, 14, 24, 42, 69, 113, 178, 276, 420, 630, 930, 1360, 1963, 2804, 3969, 5568, 7746, 10700, 14672, 19986, 27060, 36423, 48754, 64928, 86038, 113478, 149012, 194842, 253737, 329172, 425452, 547952, 703343, 899858, 1147680, 1459364, 1850310, 2339432

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

Convolution of A305123 and A000009.

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

Cf. A305102, A305101, A305104, A305122, A305105.

Cf. A116676, A305123.

nmax = 50; CoefficientList[Series[Sum[x^(2*k-1)/(1+x^(2*k-1)), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) = A305101(n) - A305122(n).

a(n) ~ exp(sqrt(n)*Pi) * log(2) / (8*Pi*sqrt(n)).

A305101 G.f.: Sum_{k>=1} x^k/(1+x^k) * Product_{k>=1} (1+x^k)/(1-x^k).

Original entry on oeis.org

0, 1, 2, 6, 11, 22, 40, 70, 116, 191, 304, 474, 726, 1094, 1624, 2384, 3453, 4950, 7030, 9890, 13798, 19108, 26264, 35858, 48652, 65615, 87996, 117396, 155826, 205854, 270728, 354506, 462306, 600544, 777184, 1002180, 1287889, 1649578, 2106152, 2680924

Offset: 0

Vaclav Kotesovec, May 25 2018

nonn

Convolution of A209423 and A000009.
Convolution of A015723 and A000041.
Convolution of A048272 and A015128.
a(n) is the number of overlined parts in all overpartitions of n. - Joerg Arndt, Jun 18 2020

Cf. A006128, A015723, A209423, A305082, A305102.

Cf. A335651 and A335666.

nmax = 40; CoefficientList[Series[Sum[x^k/(1+x^k), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

my(N=44, q='q+O('q^N)); Vec( prod(k=1,N, (1+q^k)/(1-q^k)) * sum(k=1,N, 1*q^k/(1+q^k)) ) \\ Joerg Arndt, Jun 18 2020

a(n) ~ exp(sqrt(n)*Pi) * log(2) / (4*Pi*sqrt(n)).

a(n) = A305122(n) + A305124(n).

A305104 G.f.: Sum_{k>=1} x^(2k)/(1-x^(2k)) * Product_{k>=1} (1+x^k)/(1-x^k).

Original entry on oeis.org

0, 0, 1, 2, 6, 12, 24, 44, 79, 134, 222, 358, 566, 876, 1334, 2000, 2960, 4326, 6253, 8946, 12680, 17816, 24832, 34352, 47192, 64404, 87354, 117796, 157976, 210764, 279812, 369744, 486413, 637188, 831324, 1080420, 1398968, 1805012, 2320992, 2974728, 3800618

Offset: 0

Vaclav Kotesovec, May 25 2018

nonn

Convolution A066898 of and A000009.

Convolution A090867 of and A000041.

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

Cf. A015128, A066898, A090867, A305102, A305105.

nmax = 50; CoefficientList[Series[Sum[x^(2*k)/(1-x^(2*k)), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (2*gamma + log(n/Pi^2)) * exp(Pi*sqrt(n)) / (16*Pi*sqrt(n)), where gamma is the Euler-Mascheroni constant A001620.

A305105 G.f.: Sum_{k>=1} x^(2k-1)/(1-x^(2k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

Original entry on oeis.org

0, 1, 3, 8, 17, 34, 64, 114, 195, 325, 526, 832, 1292, 1970, 2958, 4384, 6413, 9276, 13283, 18836, 26478, 36924, 51096, 70210, 95844, 130019, 175350, 235192, 313802, 416618, 550540, 724250, 948719, 1237732, 1608508, 2082600, 2686857, 3454590, 4427144, 5655652

Offset: 0

Vaclav Kotesovec, May 25 2018

Convolution of A066897 and A000009.
Convolution of A067588 and A000041.
Let A(x) = Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1)) * Product_{k >= 1} (1 + x^k)/(1 - x^k). Then A(x) = Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1)) * Product_{k >= 1} (1 + x^k)/(1 + x^k - 2*x^k) == Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1)) (mod 2). It follows from the comment in A001227 by Juri-Stepan Gerasimov, dated Jul 17 2016, that a(n) is odd iff n is a square or twice a square. - Peter Bala, Jan 10 2025

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

Cf. A001227, A015128, A066897, A067588, A305102, A305104.

nmax = 50; CoefficientList[Series[Sum[x^(2*k-1)/(1-x^(2*k-1)), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (2*gamma + log(16*n/Pi^2)) * exp(Pi*sqrt(n)) / (16*Pi*sqrt(n)), where gamma is the Euler-Mascheroni constant A001620.

A335651 a(n) is the sum, over all overpartitions of n, of the non-overlined parts.

Original entry on oeis.org

1, 5, 14, 35, 74, 150, 280, 505, 875, 1470, 2402, 3850, 6034, 9300, 14120, 21131, 31220, 45619, 65930, 94374, 133892, 188350, 262904, 364350, 501459, 685762, 932200, 1259944, 1693750, 2265380, 3015152, 3994585, 5268988, 6920700, 9053748, 11798873, 15319610, 19820738, 25557560

Offset: 1

Jeremy Lovejoy, Jun 17 2020

nonn

			The 8 overpartitions of 3 are [3], [3'], [2,1], [2,1'], [2',1], [2',1'], [1,1,1], [1',1,1], and so a(3) = 14.

K. Bringmann, J. Lovejoy, and R. Osburn, Rank and crank moments for overpartitions, Journal of Number Theory, 129 (2009), 1758-1772.

Cf. A015128, A230441, A235790, A235792, A235793, A235798, A236000, A236001, A237047, A335666.

Cf. A305102 (number of non-overlined parts).

my(N=44, q='q+O('q^N)); Vec( prod(k=1,N, (1+q^k)/(1-q^k)) * sum(k=1,N, k*q^k/(1-q^k)) ) \\ Joerg Arndt, Jun 18 2020

G.f.: (Product_{k>=1} (1+q^k)/(1-q^k)) * Sum_{n>=1} n*q^n/(1-q^n).

a(n) = A235793(n) - A335666(n). - Omar E. Pol, Jun 17 2020

A305119 G.f.: Sum_{k>=1} x^k/(1-x^k) * Product_{k>=1} 1/(1-x^k)^2.

Original entry on oeis.org

0, 1, 4, 11, 27, 58, 119, 227, 420, 744, 1287, 2160, 3561, 5739, 9113, 14224, 21924, 33327, 50126, 74531, 109802, 160211, 231875, 332821, 474313, 671072, 943411, 1317826, 1830290, 2527583, 3472446, 4746093, 6456291, 8741999, 11785768, 15822047, 21156278

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

Convolution of A006128 and A000041.

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

Cf. A000041, A000712, A006128, A305082, A305102, A305120.

nmax = 40; CoefficientList[Series[Sum[x^k/(1-x^k), {k, 1, nmax}] * Product[1/(1-x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(2*Pi*sqrt(n/3)) * (2*gamma + log(3*n/Pi^2)) / (8*3^(1/4)*Pi*n^(3/4)), where gamma is the Euler-Mascheroni constant A001620.

cp's OEIS Frontend

A305122 G.f.: Sum_{k>=1} x^(2*k)/(1+x^(2*k)) * Product_{k>=1} (1+x^k)/(1-x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A305124 G.f.: Sum_{k>=1} x^(2*k-1)/(1+x^(2*k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A305101 G.f.: Sum_{k>=1} x^k/(1+x^k) * Product_{k>=1} (1+x^k)/(1-x^k).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

A305104 G.f.: Sum_{k>=1} x^(2*k)/(1-x^(2*k)) * Product_{k>=1} (1+x^k)/(1-x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A305105 G.f.: Sum_{k>=1} x^(2*k-1)/(1-x^(2*k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A335651 a(n) is the sum, over all overpartitions of n, of the non-overlined parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A305119 G.f.: Sum_{k>=1} x^k/(1-x^k) * Product_{k>=1} 1/(1-x^k)^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A305122 G.f.: Sum_{k>=1} x^(2k)/(1+x^(2k)) * Product_{k>=1} (1+x^k)/(1-x^k).

A305124 G.f.: Sum_{k>=1} x^(2k-1)/(1+x^(2k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

A305104 G.f.: Sum_{k>=1} x^(2k)/(1-x^(2k)) * Product_{k>=1} (1+x^k)/(1-x^k).

A305105 G.f.: Sum_{k>=1} x^(2k-1)/(1-x^(2k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).