|
%I #34 Feb 02 2021 21:11:08
%S 1,1,1,2,1,3,2,3,3,5,3,6,5,7,7,9,7,12,11,13,13,17,15,21,20,24,24,29,
%T 28,36,35,40,42,50,48,58,58,67,70,80,79,93,95,106,111,125,127,145,149,
%U 166,172,191,196,222,229,250,262,289,298,330,343,373,391,427,442,486
%N Coefficients of the 5th-order mock theta function chi_0(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%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 23, 25.
%H Vaclav Kotesovec, <a href="/A053262/b053262.txt">Table of n, a(n) for n = 0..10000</a> (corrected and extended previous b-file 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.: chi_0(q) = Sum_{n>=0} q^n/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n))).
%F G.f.: chi_0(q) = 1 + Sum_{n>=0} q^(2n+1)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))).
%F a(n) is the number of partitions of 5n with rank == 1 (mod 5) minus number with rank == 0 (mod 5).
%F a(n) is the number of partitions of n with unique smallest part and all other parts <= twice the smallest part.
%F a(n) is the number of partitions where the largest part is odd and all other parts are greater than half of the largest part. - _N. Sato_, Jan 21 2010
%F a(n) ~ exp(Pi*sqrt(2*n/15)) / sqrt((5 + sqrt(5))*n). - _Vaclav Kotesovec_, Jun 12 2019
%t 1+Series[Sum[q^(2n+1)/Product[1-q^k, {k, n+1, 2n+1}], {n, 0, 49}], {q, 0, 100}]
%t nmax = 100; CoefficientList[Series[1 + Sum[x^(2*k+1)/Product[1-x^j, {j, k+1, 2*k+1}], {k, 0, Floor[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, A053263, A053264, A053265, A053266, A053267.
%K nonn,easy
%O 0,4
%A _Dean Hickerson_, Dec 19 1999
|
