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%I #23 Jan 08 2024 14:29:56
%S 0,0,1,0,1,1,1,1,2,1,2,2,3,2,4,3,4,4,5,5,7,6,8,8,9,9,12,11,14,14,16,
%T 16,20,19,23,24,27,27,32,32,37,38,43,44,51,51,58,61,67,69,78,80,89,93,
%U 102,106,118,121,134,140,153,159,175,181,198,207,224,234,256,265,288
%N Coefficients of the '5th-order' mock theta function Psi(q).
%D Dean Hickerson, A proof of the mock theta conjectures, Inventiones Mathematicae, 94 (1988) 639-660
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 18, 20
%H Vaclav Kotesovec, <a href="/A053267/b053267.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)
%H George E. Andrews and Frank G. Garvan, <a href="https://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.
%F G.f.: Psi(q) = -1 + Sum_{n>=0} q^(5n^2)/((1-q^2)(1-q^3)(1-q^7)(1-q^8)...(1-q^(5n+2))).
%F a(n) ~ exp(Pi*sqrt(2*n/15)) / (5^(3/4)*sqrt(2*phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jun 12 2019
%t Series[Sum[q^(5n^2)/Product[1-q^Abs[5k+2], {k, -n, n}], {n, 0, 4}], {q, 0, 100}]-1
%t nmax = 100; CoefficientList[Series[-1 + Sum[x^(5*k^2)/ Product[1-x^Abs[5*j+2], {j, -k, k}], {k, 0, Floor[Sqrt[nmax/5]]}], {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, A053264, A053265, A053266.
%K nonn,easy
%O 0,9
%A _Dean Hickerson_, Dec 19 1999