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%I #32 Jan 31 2021 20:46:19
%S 1,1,2,1,2,2,3,2,3,3,4,4,5,4,6,5,7,7,8,8,10,9,11,11,13,13,16,15,17,18,
%T 21,20,23,23,27,27,31,31,35,35,39,41,45,45,51,51,57,59,64,66,73,74,81,
%U 83,91,93,102,104,113,117,126,130,141,144,156,162,174,178,192,198,212
%N Coefficients of the '7th-order' mock theta function F_2(q).
%C The rank of a partition is its largest part minus the number of parts.
%D Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
%D Atle Selberg, Uber die Mock-Thetafunktionen siebenter Ordnung, Arch. Math. Naturvidenskab, 41 (1938) 3-15.
%H Terry Thibault, Frank Garvan, <a href="/A053277/b053277.txt">Table of n, a(n) for n = 0..10000</a> (terms up to n = 1000 by Seiichi Manyama)
%H George E. Andrews, <a href="http://dx.doi.org/10.1090/S0002-9947-1986-0814916-2">The fifth and seventh order mock theta functions</a>, Trans. Amer. Math. Soc., 293 (1986) 113-134.
%H Dean Hickerson, <a href="https://doi.org/10.1007/BF01394280">On the seventh order mock theta functions</a>, Inventiones Mathematicae, 94 (1988) 661-677.
%F G.f.: F_2(q) = Sum_{n >= 0} q^(n(n+1))/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1)))
%F a(n) = number of partitions of 7n+2 with rank == 1 (mod 7) minus number with rank == 2 (mod 7).
%F a(n) ~ sin(3*Pi/7) * exp(Pi*sqrt(2*n/21)) / sqrt(7*n/2). - _Vaclav Kotesovec_, Jun 15 2019
%t Series[Sum[q^(n^2+n)/Product[1-q^k, {k, n+1, 2n+1}], {n, 0, 9}], {q, 0, 100}]
%t nmax = 100; CoefficientList[Series[Sum[x^(k^2+k)/Product[1-x^j, {j, k+1, 2*k+1}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 14 2019 *)
%Y Other '7th-order' mock theta functions are at A053275, A053276, A053278, A053279, A053280.
%K nonn,easy
%O 0,3
%A _Dean Hickerson_, Dec 19 1999
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