%I #25 Jan 08 2024 14:30:04
%S 0,1,0,0,1,1,0,0,0,1,1,0,1,1,0,0,1,1,0,1,1,1,1,0,1,2,1,0,1,1,1,1,1,2,
%T 1,1,2,2,1,1,2,2,1,1,2,2,2,1,2,3,2,2,3,3,2,2,2,3,3,2,3,4,2,2,4,4,3,3,
%U 4,4,4,3,4,5,4,4,5,5,4,4,5,6,5,4,6,7,5,5,6,7,6,6,7,7,7,6,8,9,7,7,9
%N Coefficients of the '5th-order' mock theta function phi_1(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%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 22, 25.
%H Vaclav Kotesovec, <a href="/A053259/b053259.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="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.
%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.: phi_1(q) = Sum_{n>=0} q^(n+1)^2 (1+q)(1+q^3)...(1+q^(2n-1)).
%F a(n) is the number of partitions of n into odd parts such that each occurs at most twice, the largest part is unique and if k occurs as a part then all smaller positive odd numbers occur.
%F a(n) ~ exp(Pi*sqrt(n/30)) / (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^(n+1)^2 Product[1+q^(2k-1), {k, 1, n}], {n, 0, 9}], {q, 0, 100}]
%t nmax = 100; CoefficientList[Series[Sum[x^((k+1)^2) * Product[1+x^(2*j-1), {j, 1, k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 12 2019 *)
%Y Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053260, A053261, A053262, A053263, A053264, A053265, A053266, A053267.
%K nonn,easy
%O 0,26
%A _Dean Hickerson_, Dec 19 1999