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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

N. J. A. Sloane

STATUS

approved

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Last modified February 24 08:02 EST 2020. Contains 332199 sequences. (Running on oeis4.)