A261389 -id:A261389 - 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

A261386 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(2*k).

Original entry on oeis.org

1, 4, 16, 56, 176, 520, 1456, 3896, 10048, 25100, 60960, 144440, 334752, 760456, 1696464, 3722224, 8043040, 17135624, 36031104, 74840568, 153680928, 312198160, 627828272, 1250540024, 2468443296, 4830809868, 9377190336, 18061370288, 34531009760, 65552873736

Offset: 0

Vaclav Kotesovec, Aug 17 2015

nonn

Convolution of A161870 and A026011.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
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. 19.

Cf. A015128, A026011, A156616, A161870, A216406, A261384, A261389.

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

a(n) ~ exp(1/6 + 3/2*(7*Zeta(3))^(1/3) * n^(2/3)) * (7*Zeta(3))^(2/9) / (A^2 * 2^(2/3) * n^(13/18) * sqrt(3*Pi)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

G.f.: exp(Sum_{k>=1} (sigma_2(2*k) - sigma_2(k))*x^k/k). - Ilya Gutkovskiy, Apr 14 2019

A261452 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(2*k-1).

Original entry on oeis.org

1, 2, 8, 24, 66, 176, 448, 1096, 2608, 6042, 13664, 30280, 65856, 140800, 296432, 615264, 1260306, 2550368, 5102616, 10101000, 19797344, 38439088, 73976160, 141179480, 267300752, 502283714, 937077808, 1736296304, 3196144032, 5846632656, 10631038400

Offset: 0

Vaclav Kotesovec, Aug 19 2015

nonn

Convolution of A253289 and A255835.

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, p. 26.

Cf. A156616, A261386, A005309, A261451, A261389.

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

a(n) ~ 2^(1/3) * (7*Zeta(3))^(1/18) * exp(1/6 - Pi^4/(672*Zeta(3)) - Pi^2 * n^(1/3)/(4*(7*Zeta(3))^(1/3)) + 3/2*(7*Zeta(3))^(1/3) * n^(2/3)) / (A^2 * sqrt(3) * n^(5/9)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A261451 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(k+1).

Original entry on oeis.org

1, 4, 14, 44, 124, 328, 824, 1980, 4590, 10320, 22584, 48268, 101016, 207432, 418704, 832032, 1629764, 3150280, 6014998, 11354084, 21204488, 39206168, 71811256, 130369900, 234704360, 419195412, 743085912, 1307823672, 2286094704, 3970174648, 6852048368

Offset: 0

Vaclav Kotesovec, Aug 19 2015

nonn

Convolution of A005380 and A219555.

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

Cf. A156616, A005309, A261386, A261452, A261389.

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

a(n) ~ (7*Zeta(3))^(13/36) * exp(1/12 - Pi^4/(336*Zeta(3)) + Pi^2 * n^(1/3) / (2^(5/3) * (7*Zeta(3))^(1/3)) + 3/2 * ((7*Zeta(3))/2)^(1/3) * n^(2/3)) / (A * 2^(35/18) * 3^(1/2) * Pi * n^(31/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A005309 Fermionic string states.

Original entry on oeis.org

1, 0, 2, 4, 8, 16, 32, 60, 114, 212, 384, 692, 1232, 2160, 3760, 6480, 11056, 18728, 31474, 52492, 86976, 143176, 234224, 380988, 616288, 991624, 1587600, 2529560, 4011808, 6334656, 9960080, 15596532, 24327122, 37801568, 58525152, 90291232, 138825416

Offset: 0

N. J. A. Sloane

nonn

See the reference for precise definition.

The g.f. -(1-2*z+2*z**2)/(-1+2*z) conjectured by Simon Plouffe in his 1992 dissertation is not correct.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
T. Curtright, Counting symmetry patterns in the spectra of strings, in H. J. de Vega and N. Sánchez, editors, String Theory, Quantum Cosmology and Quantum Gravity. Integrable and Conformal Invariant Theories. World Scientific, Singapore, 1987, pp. 304-333, eq. (3.39) and Table 3.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Cf. A156616, A261451, A261386, A261452, A261389.

G.f. Product_{k>=1} ((1+x^k)/(1-x^k))^(k-1). - Vaclav Kotesovec, Aug 19 2015
Convolution of A052847 and A052812. - Vaclav Kotesovec, Aug 19 2015
a(n) ~ 2^(7/18) * (7*Zeta(3))^(1/36) * exp(1/12 - Pi^4/(336*Zeta(3)) - Pi^2 * n^(1/3) / (2^(5/3)*(7*Zeta(3))^(1/3)) + 3/2 * (7*Zeta(3)/2)^(1/3) * n^(2/3)) / (A * sqrt(3) * n^(19/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 19 2015

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

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A261386 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(2*k).

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A261452 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(2*k-1).

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A261451 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(k+1).

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A005309 Fermionic string states.

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