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%I #8 Jan 11 2017 08:07:47
%S 1,1,2,3,5,7,11,15,20,28,38,50,67,87,113,146,187,237,301,378,473,590,
%T 732,903,1113,1364,1666,2030,2464,2981,3600,4332,5201,6229,7442,8869,
%U 10551,12521,14829,17531,20684,24357,28638,33607,39375,46062,53798,62736
%N Expansion of Product_{k>=1} (1 - x^(8*(2*k-1))) * (1 - x^(8*k)) / (1 - x^k).
%C In general, if r>=2 and g.f. = Product_{k>=1} (1-x^(r*(2*k-1))) * (1-x^(r*k)) / (1-x^k), then
%C a(n, r) ~ 2*Pi * BesselI(1, Pi/6 * sqrt((24*n-1)*(2*r-3)/(2*r))) / (r*sqrt((24*n-1)/(2*r-3))).
%C a(n, r) ~ exp(Pi * sqrt((2/3 - 1/r)*n)) * (2*r-3)^(1/4) / (2 * 3^(1/4) * r^(3/4) * n^(3/4)) * (1 -(3*sqrt(3*r)/(8*Pi*sqrt(2*r-3)) + Pi*sqrt(2*r-3)/(48*sqrt(3*r))) / sqrt(n) + (Pi^2*(2*r-3)/(13824*r) - 45*r/(128*Pi^2*(2*r-3)) + 5/128)/n).
%D D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.
%H Vaclav Kotesovec, <a href="/A280938/b280938.txt">Table of n, a(n) for n = 0..5000</a>
%H Andrew Sills, <a href="https://works.bepress.com/andrew_sills/40/">Rademacher-Type Formulas for Restricted Partition and Overpartition Functions</a>, Ramanujan Journal, 23 (1-3): 253-264, 2010.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bailey_pair">Bailey pair</a>.
%F a(n) ~ Pi * BesselI(1, Pi * sqrt(13*(24*n-1))/24) / (4*sqrt((24*n-1)/13)).
%F a(n) ~ exp(Pi*sqrt(13*n/6)/2) * 13^(1/4) / (2^(13/4) * 3^(1/4) * n^(3/4)) * (1 -(3*sqrt(3)/(2*Pi*sqrt(26)) + Pi*sqrt(13)/(96*sqrt(6)))/sqrt(n) + (13*Pi^2/110592 - 45/(208*Pi^2) + 5/128)/n).
%t nmax = 50; CoefficientList[Series[Product[(1-x^(8*(2*k-1))) * (1-x^(8*k)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A000700 (r=2), A070047 (r=3), A108961 (r=4), A108962 (r=5), A271661 (r=6), A280937 (r=7).
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, Jan 11 2017
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