A285460 -id:A285460 - cp's OEIS frontend

A156616 G.f.: Product_{n>0} ((1+x^n)/(1-x^n))^n.

1, 2, 6, 16, 38, 88, 196, 420, 878, 1794, 3584, 7032, 13572, 25792, 48352, 89512, 163774, 296444, 531234, 943072, 1659560, 2896376, 5015700, 8622108, 14718652, 24960138, 42062200, 70458160, 117349856, 194381704, 320295312, 525123604

Offset: 0

R. J. Mathar, Feb 11 2009

Generating function for a sum over strict plane partitions weighted with 2 powered to their number of connected components.
The inverse Euler transform is apparently 2, 3, 6, 6, 10, 9, 14, 12, 18, 15, 22, 18, 26, 21, ..., A016825 interlaced with A008585. - R. J. Mathar, Apr 23 2009
In general, for m >= 1, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(m*k), then a(n) ~ exp(m/12 + 3/2 * (7*m*Zeta(3)/2)^(1/3) * n^(2/3)) * m^(1/6 + m/36) * (7*Zeta(3))^(1/6 + m/36) / (A^m * 2^(2/3 + m/9) * sqrt(3*Pi) * n^(2/3 + m/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 17 2015
In general, for m >= 0, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(k^m), then a(n) ~ ((2^(m+2)-1) * Gamma(m+2) * Zeta(m+2) / (2^(2*m+3) * n))^((1-2*Zeta(-m))/(2*m+4)) * exp((m+2)/(m+1) * ((2^(m+2)-1) * n^(m+1) * Gamma(m+2) * Zeta(m+2) / 2^(m+1))^(1/(m+2)) + Zeta'(-m)) / sqrt((m+2)*Pi*n). - Vaclav Kotesovec, Aug 19 2015

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
Ali H. Al-Saedi, Congruences for restricted plane overpartitions modulo 4 and 8, Raman. J. 48 (2) (2019) 251
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, p. 18.
Mirjana Vuletic, A generalization of MacMahon's formula, Trans. Am. Math. Soc. 361 (2009) 2789-2804.

Cf. A000219, A015128, A026007, A261384, A261386, A261389.
Cf. A206622, A206623, A206624, A261519, A261520.
Cf. A285462, A285447, A285460, A285461.
Cf. A285446, A285458, A285459.
Cf. A306081.

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 17 2015 *)

{a(n)=polcoeff(exp(sum(m=1,n,(sigma(2*m,2)-sigma(m,2))/2*x^m/m)+x*O(x^n)),n)} \\ Paul D. Hanna, May 01 2010

Convolve A000219 with A026007.
O.g.f.: exp( Sum_{n>=1} (sigma_2(2n) - sigma_2(n))/2 *x^n/n ), where sigma_2(n) is the sum of squares of divisors of n (A001157). - Paul D. Hanna, May 01 2010
a(n) ~ exp(1/12 + 3 * 2^(-4/3) * (7*Zeta(3))^(1/3) * n^(2/3)) * (7*Zeta(3))^(7/36) / (A * 2^(7/9) * sqrt(3*Pi) * n^(25/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 17 2015
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A076577(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 30 2017
G.f.: A(x) = exp( 2*Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 - x^(2*n+1))^2) ). Cf. A000122 and A302237. - Peter Bala, Dec 23 2021

A285447 Expansion of Product_{k>=1} ((1 + x^(3*k)) / (1 - x^k))^k.

Original entry on oeis.org

1, 1, 3, 7, 14, 27, 56, 101, 190, 347, 617, 1082, 1895, 3230, 5490, 9226, 15332, 25259, 41356, 67021, 107989, 172789, 274613, 433815, 681650, 1064661, 1654739, 2559029, 3938438, 6033967, 9205152, 13982675, 21156174, 31886290, 47879210, 71636483, 106814323

Offset: 0

Vaclav Kotesovec, Apr 19 2017

nonn

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

Cf. A266648, A285446.

Cf. A156616, A285462, A285460, A285461.

nmax = 40; CoefficientList[Series[Product[((1+x^(3*k))/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(1/12 + 2^(-4/3) * 3^(2/3) * (13*Zeta(3))^(1/3) * n^(2/3)) * (13*Zeta(3))^(7/36) / (A * 2^(7/9) * 3^(25/36) * sqrt(Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.

A285461 Expansion of Product_{k>=1} ((1 + x^(5*k)) / (1 - x^k))^k.

Original entry on oeis.org

1, 1, 3, 6, 13, 25, 49, 89, 166, 295, 526, 909, 1571, 2657, 4475, 7432, 12257, 20000, 32436, 52126, 83285, 132057, 208221, 326202, 508372, 787777, 1214828, 1863932, 2847020, 4328765, 6554359, 9882795, 14843999, 22210386, 33112817, 49192218, 72834243

Offset: 0

Vaclav Kotesovec, Apr 19 2017

nonn

In general, if m >= 1 and g.f. = Product_{k>=1} ((1 + x^(m*k)) / (1 - x^k))^k, then a(n, m) ~ exp(1/12 + 3 * 2^(-4/3) * (4 + 3/m^2)^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * (4 + 3/m^2)^(7/36) * Zeta(3)^(7/36) / (A * 2^(7/9) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.

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

Cf. A156616 (m=1), A285462 (m=2), A285447 (m=3), A285460 (m=4).

Cf. A024789.

nmax = 40; CoefficientList[Series[Product[((1+x^(5*k))/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(1/12 + 3 * 2^(-4/3) * 5^(-2/3) * (103*Zeta(3))^(1/3) * n^(2/3)) * (103*Zeta(3))^(7/36) / (A * 2^(7/9) * 5^(7/18) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.

A285462 Expansion of Product_{k>=1} ((1 + x^(2*k)) / (1 - x^k))^k.

Original entry on oeis.org

1, 1, 4, 7, 18, 32, 72, 127, 257, 454, 861, 1497, 2719, 4654, 8171, 13781, 23564, 39159, 65559, 107455, 176712, 286000, 463200, 740910, 1184123, 1873656, 2959376, 4636145, 7245680, 11246590, 17409731, 26792371, 41114202, 62769820, 95553779, 144803917

Offset: 0

Vaclav Kotesovec, Apr 19 2017

nonn

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

Cf. A156616, A285447, A285460, A285461.

Cf. A279328.

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

a(n) ~ exp(1/12 + 3 * (19*Zeta(3))^(1/3) * n^(2/3) / 4) * (19*Zeta(3))^(7/36) / (A * 2^(7/6) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.

A285472 Expansion of Product_{k>=1} ((1 + x^(4*k)) / (1 - x^k)).

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 13, 18, 28, 38, 55, 74, 106, 140, 192, 253, 342, 444, 588, 758, 992, 1267, 1634, 2072, 2650, 3334, 4218, 5276, 6627, 8234, 10262, 12682, 15708, 19308, 23764, 29070, 35597, 43340, 52792, 64008, 77622, 93724, 113160, 136124, 163712, 196225

Offset: 0

Vaclav Kotesovec, Apr 19 2017

nonn

a(n) is the number of overpartitions wherein only parts that are a multiple of four may be overlined. - Alois P. Heinz, Feb 03 2025

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A006950, A015128, A279328, A266648, A285460.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(irem(i, 4)=0, 2, 1)*add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..45);  # Alois P. Heinz, Feb 03 2025

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

a(n) ~ sqrt(3) * exp(sqrt(3*n)*Pi/2) / (16*n).

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A156616 G.f.: Product_{n>0} ((1+x^n)/(1-x^n))^n.

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A285447 Expansion of Product_{k>=1} ((1 + x^(3*k)) / (1 - x^k))^k.

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A285461 Expansion of Product_{k>=1} ((1 + x^(5*k)) / (1 - x^k))^k.

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A285462 Expansion of Product_{k>=1} ((1 + x^(2*k)) / (1 - x^k))^k.

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A285472 Expansion of Product_{k>=1} ((1 + x^(4*k)) / (1 - x^k)).

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