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A280937 Expansion of Product_{k>=1} ((1 - x^(7*(2*k-1))) * (1 - x^(7*k)) / (1 - x^k)). 6

%I

%S 1,1,2,3,5,7,11,13,20,26,36,46,63,79,105,132,171,213,273,336,425,522,

%T 650,793,981,1188,1456,1756,2136,2563,3098,3698,4443,5285,6312,7477,

%U 8891,10489,12415,14599,17206,20165,23678,27659,32363,37698,43958,51058,59361

%N Expansion of Product_{k>=1} ((1 - x^(7*(2*k-1))) * (1 - x^(7*k)) / (1 - x^k)).

%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="/A280937/b280937.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) ~ 2*Pi * BesselI(1, Pi/6 * sqrt(11*(24*n-1)/14)) / (7*sqrt((24*n-1)/11)).

%F a(n) ~ exp(Pi * sqrt(11*n/21)) * 11^(1/4) / (2 * 3^(1/4) * 7^(3/4) * n^(3/4)) * (1 -(3*sqrt(21)/(8*Pi*sqrt(11)) + Pi*sqrt(11)/(48*sqrt(21)))/sqrt(n) + (11*Pi^2/96768 - 315/(1408*Pi^2) + 5/128)/n).

%t nmax = 50; CoefficientList[Series[Product[(1-x^(7*(2*k-1))) * (1-x^(7*k)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A000700, A070047, A108961, A108962, A271661, A280938.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Jan 11 2017

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Last modified August 2 02:53 EDT 2021. Contains 346409 sequences. (Running on oeis4.)