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%I #30 Jan 08 2024 14:30:08
%S 1,0,1,-1,1,-1,2,-2,1,-1,2,-2,2,-2,2,-3,3,-2,3,-4,4,-4,4,-5,5,-4,5,-6,
%T 6,-6,7,-8,7,-7,8,-9,10,-9,10,-12,11,-11,13,-14,14,-15,16,-17,17,-16,
%U 19,-21,20,-21,23,-25,25,-25,27,-29,30,-30,32,-35,35,-35,39,-41,41,-43,45,-49,50,-49,53,-57,58,-59,63,-67,68
%N Coefficients of the '5th-order' mock theta function f_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%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 22.
%H Vaclav Kotesovec, <a href="/A053257/b053257.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 Dean Hickerson, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00210590X">A proof of the mock theta conjectures</a>, Inventiones Mathematicae, 94 (1988) 639-660.
%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_1(q) = Sum_{n>=0} q^(n^2+n)/((1+q)(1+q^2)...(1+q^n)).
%F Consider partitions of n into parts differing by at least 2 and with smallest part at least 2. a(n) is the number of them with largest part even minus number with largest part odd.
%F a(n) ~ (-1)^n * sqrt(phi) * exp(Pi*sqrt(n/15)) / (2*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jun 15 2019
%t Series[Sum[q^(n^2+n)/Product[1+q^k, {k, 1, n}], {n, 0, 9}], {q, 0, 100}]
%t nmax = 100; CoefficientList[Series[Sum[x^(k^2+k) / Product[1+x^j, {j, 1, k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 15 2019 *)
%Y Other '5th-order' mock theta functions are at A053256, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053266, A053267.
%K sign,easy
%O 0,7
%A _Dean Hickerson_, Dec 19 1999