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 A005308 Bosonic string states. (Formerly M0310) 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS See the reference for precise definition. 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. Plouffe, Simon, Master's Thesis, copy at the arXiv site 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 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 *) CROSSREFS Cf. A003293, A005986. Sequence in context: A222710 A032278 A222738 * A151532 A056503 A256217 Adjacent sequences:  A005305 A005306 A005307 * A005309 A005310 A005311 KEYWORD nonn AUTHOR STATUS approved

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Last modified August 3 02:44 EDT 2021. Contains 346435 sequences. (Running on oeis4.)