A261451 -id:A261451 - cp's OEIS frontend

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

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.

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

A300412 a(n) = [x^n] Product_{k>=1} ((1 + nx^k)/(1 - nx^k))^k.

Original entry on oeis.org

1, 2, 16, 144, 1376, 15800, 210816, 3333372, 61688448, 1318588146, 32004369200, 869282342632, 26099925704928, 857736429098848, 30605729417479104, 1177841009504482200, 48614265201514729984, 2141639401723095243324, 100282931820560447963568, 4973060138191518242569120

Offset: 0

Ilya Gutkovskiy, Mar 05 2018

nonn

			The table of coefficients of x^k in expansion of Product_{k>=1} ((1 + n*x^k)/(1 - n*x^k))^k begins:
n = 0: (1),  0,   0,    0,     0,       0,  ...
n = 1:  1,  (2),  6,   16,    38,      88,  ...
n = 2:  1,   4, (16),  60,   192,     596,  ...
n = 3:  1,   6,  30, (144),  582,    2280,  ...
n = 4:  1,   8,  48,  280, (1376),   6568,  ...
n = 5:  1,  10,  70,  480,  2790,  (15800), ...

Cf. A156616, A261563, A266942, A270919, A270924, A292419, A298985, A298987.

Table[SeriesCoefficient[Product[((1 + n x^k)/(1 - n x^k))^k, {k, 1, n}], {x, 0, n}], {n, 0, 19}]

a(n) ~ 2 * n^n * (1 + 4/n + 14/n^2 + 44/n^3 + 124/n^4 + 328/n^5 + 824/n^6 + 1980/n^7 + 4590/n^8 + 10320/n^9 + 22584/n^10 + ...), for coefficients see A261451. - Vaclav Kotesovec, Mar 05 2018

cp's OEIS Frontend

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

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A005309 Fermionic string states.

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A300412 a(n) = [x^n] Product_{k>=1} ((1 + nx^k)/(1 - nx^k))^k.

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cp's OEIS Frontend

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

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Author

Keywords

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A005309 Fermionic string states.

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Author

Keywords

Comments

References

Links

Crossrefs

Formula

A300412 a(n) = [x^n] Product_{k>=1} ((1 + n*x^k)/(1 - n*x^k))^k.

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Mathematica

Formula

A300412 a(n) = [x^n] Product_{k>=1} ((1 + nx^k)/(1 - nx^k))^k.