A005308 Bosonic string states.
1, 0, 0, 0, 1, 1, 2, 2, 4, 4, 7, 8, 14, 16, 25, 31, 47, 58, 85, 107, 153, 195, 271, 348, 480, 616, 834, 1077, 1445, 1863, 2478, 3194, 4216, 5431, 7118, 9157, 11942, 15329, 19884, 25485, 32916, 42090, 54147, 69093, 88563, 112769, 144056, 183028, 233112, 295525
Offset: 1
Keywords
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
- 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.
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Programs
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Mathematica
nmax = 50; Rest[CoefficientList[Series[x/(1-x)*Product[1/(1-x^k)^((2*k - 5 + (-1)^k)/4), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 10 2016 *)
Formula
G.f.: Product (1 - x^k)^{-c(k)}; c(k) = 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, ....
Euler transform gives sequence with g.f. = x^3/((x+1)*(x-1)^2), Simon Plouffe, Master's Thesis, UQAM 1992.
a(n) ~ 2^(1/4) * exp(1/24 - 25*Pi^4/(3456*Zeta(3)) - 5*Pi^2 * n^(1/3) / (24*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2) / (A^(1/2) * sqrt(3) * Zeta(3)^(23/72) * n^(13/72)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Sep 26 2016
Comments