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%I #23 Feb 02 2021 22:00:29
%S 1,0,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,5,6,6,7,8,8,10,11,11,13,14,15,17,
%T 18,19,22,24,25,28,30,32,36,39,41,45,49,52,57,61,65,71,76,81,88,94,
%U 100,109,116,123,133,142,151,163,174,184,198,211,224,240,255,271,290
%N Coefficients of the '5th-order' mock theta function F_0(q).
%D Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 22, 23, 25.
%H Vaclav Kotesovec, <a href="/A053264/b053264.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)
%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 George E. Andrews and Frank G. Garvan, <a href="http://dx.doi.org/10.1016/0001-8708(89)90070-4">Ramanujan's "lost" notebook VI: The mock theta conjectures</a>, Advances in Mathematics, 73 (1989) 242-255.
%H George N. Watson, <a href="https://doi.org/10.1112/plms/s2-42.1.274">The mock theta functions (2)</a>, Proc. London Math. Soc., series 2, 42 (1937) 274-304.
%F G.f.: F_0(q) = Sum_{n>=0} q^(2n^2)/((1-q)(1-q^3)...(1-q^(2n-1))).
%F a(n) is the number of partitions of n into odd parts, each of which occurs at least twice, such that if k occurs then all smaller positive odd numbers occur.
%F a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(3/2)*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jun 12 2019
%t Series[Sum[q^(2n^2)/Product[1-q^(2k+1), {k, 0, n-1}], {n, 0, 7}], {q, 0, 100}]
%t nmax = 100; CoefficientList[Series[Sum[x^(2*k^2) / Product[1-x^(2*j+1), {j, 0, k-1}], {k, 0, Floor[Sqrt[nmax/2]]}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 12 2019 *)
%Y Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053265, A053266, A053267.
%K nonn,easy
%O 0,9
%A _Dean Hickerson_, Dec 19 1999
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