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%I A053263 #41 Jan 08 2024 14:30:01
%S A053263 1,2,2,3,3,4,4,6,5,7,8,9,9,12,12,15,15,18,19,23,23,27,30,33,34,41,42,
%T A053263 49,51,57,61,69,72,81,87,96,100,113,119,132,140,153,163,180,188,208,
%U A053263 221,240,253,278,294,319,339,366,388,422,443,481,510,549,580,626,662
%N A053263 Coefficients of the '5th-order' mock theta function chi_1(q).
%C A053263 The rank of a partition is its largest part minus the number of parts.
%C A053263 Number of partitions of n such that 2*(least part) > greatest part. - _Clark Kimberling_, Feb 16 2014
%C A053263 Also the number of partitions of n with the same median as maximum. These are conjugate to the partitions described above. For minimum instead of maximum we have A361860. - _Gus Wiseman_, Apr 23 2023
%D A053263 Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355
%D A053263 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 25
%H A053263 Vaclav Kotesovec, <a href="/A053263/b053263.txt">Table of n, a(n) for n = 0..5000</a> (terms 0..1000 from Seiichi Manyama)
%H A053263 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 A053263 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 A053263 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 A053263 G.f.: chi_1(q) = Sum_{n>=0} q^n/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))).
%F A053263 G.f.: chi_1(q) = 1 + Sum_{n>=0} q^(2n+1) (1+q^n)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))).
%F A053263 a(n) is twice the number of partitions of 5n+3 with rank == 2 (mod 5) minus number with rank == 0 or 1 (mod 5).
%F A053263 a(n) - 1 is the number of partitions of n with unique smallest part and all other parts <= one plus twice the smallest part.
%F A053263 a(n) ~ sqrt(phi/2) * exp(Pi*sqrt(2*n/15)) / (5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jun 16 2019
%e A053263 From _Gus Wiseman_, Apr 20 2023: (Start)
%e A053263 The a(1) = 1 through a(8) = 6 partitions such that 2*(minimum) > (maximum):
%e A053263   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A053263        (11)  (111)  (22)    (32)     (33)      (43)       (44)
%e A053263                     (1111)  (11111)  (222)     (322)      (53)
%e A053263                                      (111111)  (1111111)  (332)
%e A053263                                                           (2222)
%e A053263                                                           (11111111)
%e A053263 The a(1) = 1 through a(8) = 6 partitions such that (median) = (maximum):
%e A053263   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A053263        (11)  (111)  (22)    (221)    (33)      (331)      (44)
%e A053263                     (1111)  (11111)  (222)     (2221)     (332)
%e A053263                                      (111111)  (1111111)  (2222)
%e A053263                                                           (22211)
%e A053263                                                           (11111111)
%e A053263 (End)
%t A053263 1+Series[Sum[q^(2n+1)(1+q^n)/Product[1-q^k, {k, n+1, 2n+1}], {n, 0, 49}], {q, 0, 100}]
%t A053263 (* Also: *)
%t A053263 Table[Count[ IntegerPartitions[n], p_ /; 2 Min[p] > Max[p]], {n, 40}]
%t A053263 (* _Clark Kimberling_, Feb 16 2014 *)
%t A053263 nmax = 100; CoefficientList[Series[1 + Sum[x^(2*k+1)*(1+x^k) / 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 A053263 Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053264, A053265, A053266, A053267.
%Y A053263 Cf. A047993, A240874, A361800, A361849 (ranks A361856), A361860.
%Y A053263 A000041 counts integer partitions, strict A000009, odd-length A027193.
%Y A053263 A359893 and A359901 count partitions by median.
%Y A053263 Cf. A013580, A079309, A237753, A237757, A237800, A361848, A361858, A361859.
%K A053263 nonn,easy
%O A053263 0,2
%A A053263 _Dean Hickerson_, Dec 19 1999

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