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%I #51 Aug 12 2023 23:00:08
%S 1,2,3,4,6,8,10,14,18,22,29,36,44,56,68,82,101,122,146,176,210,248,
%T 296,350,410,484,566,660,772,896,1038,1204,1391,1602,1846,2120,2428,
%U 2784,3182,3628,4138,4708,5347,6072,6880,7784,8804,9940,11208,12630
%N Coefficients of the '3rd-order' mock theta function omega(q).
%C Empirical: a(n) is the number of integer partitions mu of 2n+1 such that the diagram of mu has an odd number of cells in each row and in each column. - _John M. Campbell_, Apr 24 2020
%C From _Gus Wiseman_, Jun 26 2022: (Start)
%C By Campbell's conjecture above that a(n) is the number of partitions of 2n+1 with all odd parts and all odd conjugate parts, the a(0) = 1 through a(5) = 8 partitions are (B = 11):
%C (1) (3) (5) (7) (9) (B)
%C (111) (311) (511) (333) (533)
%C (11111) (31111) (711) (911)
%C (1111111) (51111) (33311)
%C (3111111) (71111)
%C (111111111) (5111111)
%C (311111111)
%C (11111111111)
%C These partitions are ranked by A352143.
%C (End)
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 15, 17, 31.
%H Vaclav Kotesovec, <a href="/A053253/b053253.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Seiichi Manyama)
%H Leila A. Dragonette, <a href="http://dx.doi.org/10.1090/S0002-9947-1952-0049927-8">Some asymptotic formulas for the mock theta series of Ramanujan</a>, Trans. Amer. Math. Soc., 72 (1952) 474-500.
%H John F. R. Duncan, Michael J. Griffin and Ken Ono, <a href="http://arxiv.org/abs/1503.01472">Proof of the Umbral Moonshine Conjecture</a>, arXiv:1503.01472 [math.RT], 2015.
%H George N. Watson, <a href="https://doi.org/10.1112/jlms/s1-11.1.55">The final problem: an account of the mock theta functions</a>, J. London Math. Soc., 11 (1936) 55-80.
%F G.f.: omega(q) = Sum_{n>=0} q^(2*n*(n+1))/((1-q)*(1-q^3)*...*(1-q^(2*n+1)))^2.
%F G.f.: Sum_{k>=0} x^k/((1-x)(1-x^3)...(1-x^(2k+1))). - _Michael Somos_, Aug 18 2006
%F G.f.: (1 - G(0))/(1-x) where G(k) = 1 - 1/(1-x^(2*k+1))/(1-x/(x-1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jan 18 2013
%F a(n) ~ exp(Pi*sqrt(n/3)) / (4*sqrt(n)). - _Vaclav Kotesovec_, Jun 10 2019
%t Series[Sum[q^(2n(n+1))/Product[1-q^(2k+1), {k, 0, n}]^2, {n, 0, 6}], {q, 0, 100}]
%o (PARI) {a(n)=local(A); if(n<0, 0, A=1+x*O(x^n); polcoeff( sum(k=0, (sqrtint(2*n+1)-1)\2, A*=(x^(4*k)/(1-x^(2*k+1))^2 +x*O(x^(n-2*(k^2-k))))), n))} /* _Michael Somos_, Aug 18 2006 */
%o (PARI) {a(n)=local(A); if(n<0, 0, n++; A=1+x*O(x^n); polcoeff( sum(k=0, n-1, A*=(x/(1-x^(2*k+1)) +x*O(x^(n-k)))), n))} /* _Michael Somos_, Aug 18 2006 */
%Y Other '3rd-order' mock theta functions are at A000025, A053250, A053251, A053252, A053254, A053255, A261401.
%Y Cf. A095913(n)=a(n-3).
%Y Cf. A259094.
%Y Conjectured to count the partitions ranked by A352143.
%Y A069911 = strict partitions w/ all odd parts, ranked by A258116.
%Y A078408 = partitions w/ all odd parts, ranked by A066208.
%Y A117958 = partitions w/ all odd parts and multiplicities, ranked by A352142.
%Y Cf. A000009, A000290, A000701, A035363 (complement A086543), A035444, A035457, A045931, A055922, A258117.
%K nonn,easy
%O 0,2
%A _Dean Hickerson_, Dec 19 1999
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