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A005308 Bosonic string states.

%I A005308 M0310 #46 Dec 28 2024 10:19:53
%S A005308 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,
%T A005308 616,834,1077,1445,1863,2478,3194,4216,5431,7118,9157,11942,15329,
%U A005308 19884,25485,32916,42090,54147,69093,88563,112769,144056,183028,233112,295525
%N A005308 Bosonic string states.
%C A005308 See the reference for precise definition.
%D A005308 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005308 Vaclav Kotesovec, <a href="/A005308/b005308.txt">Table of n, a(n) for n = 1..10000</a>
%H A005308 T. Curtright, <a href="http://inspirehep.net/record/234762/files/CountingStringStates.pdf">Counting symmetry patterns in the spectra of strings</a>, 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.
%H A005308 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%F A005308 G.f.: Product (1 - x^k)^{-c(k)}; c(k) = 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, ....
%F A005308 Euler transform gives sequence with g.f. = x^3/((x+1)*(x-1)^2), _Simon Plouffe_, Master's Thesis, UQAM 1992.
%F A005308 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
%t A005308 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 *)
%Y A005308 Cf. A003293, A005986.
%K A005308 nonn
%O A005308 1,7
%A A005308 _N. J. A. Sloane_