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%I #48 Mar 06 2020 22:38:13
%S 0,1,1,2,2,2,4,4,4,6,7,8,10,11,12,16,18,20,24,26,30,36,40,44,52,58,64,
%T 74,82,91,104,116,128,144,159,176,198,218,240,268,294,324,360,394,432,
%U 478,524,572,630,688,752,826,900,980,1072,1168,1270,1386,1505,1634
%N Coefficients of the '10th-order' mock theta function psi(q).
%C Number of partitions (d1,d2,...,dm) of n such that 0 < d1/1 <= d2/2 <= ... <= dm/m. - _Seiichi Manyama_, Mar 17 2018
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 9
%H Vaclav Kotesovec, <a href="/A053282/b053282.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Seiichi Manyama)
%H Youn-Seo Choi, <a href="http://dx.doi.org/10.1007/s002220050318">Tenth order mock theta functions in Ramanujan's lost notebook</a>, Inventiones Mathematicae, 136 (1999) p. 497-569.
%H Michele Nardelli, Antonio Nardelli, <a href="https://pdfs.semanticscholar.org/e28c/47202ef1caaafd703fda86d278a8b6771111.pdf">On the Ramanujan's Mock theta functions of tenth order: new possible mathematical developments and mathematical connections with some sectors of Particle Physics and Black Hole physics II</a>, Università degli Studi di Napoli (Italy, 2019).
%F G.f.: psi(q) = Sum_{n >= 0} q^((n+1)(n+2)/2)/((1-q)(1-q^3)...(1-q^(2n+1))).
%F a(n) ~ exp(Pi*sqrt(n/5)) / (2*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jun 12 2019
%e From _Seiichi Manyama_, Mar 17 2018: (Start)
%e n | Partition (d1,d2,...,dm) | (d1/1, d2/2, ... , dm/m)
%e --+--------------------------+-------------------------
%e 1 | (1) | (1)
%e 2 | (2) | (2)
%e 3 | (3) | (3)
%e | (1, 2) | (1, 1)
%e 4 | (4) | (4)
%e | (1, 3) | (1, 3/2)
%e 5 | (5) | (5)
%e | (1, 4) | (1, 2)
%e 6 | (6) | (6)
%e | (1, 5) | (1, 5/2)
%e | (2, 4) | (2, 2)
%e | (1, 2, 3) | (1, 1, 1)
%e 7 | (7) | (7)
%e | (1, 6) | (1, 3)
%e | (2, 5) | (2, 5/2)
%e | (1, 2, 4) | (1, 1, 4/3)
%e 8 | (8) | (8)
%e | (1, 7) | (1, 7/2)
%e | (2, 6) | (2, 3)
%e | (1, 2, 5) | (1, 1, 5/3)
%e 9 | (9) | (9)
%e | (1, 8) | (1, 4)
%e | (2, 7) | (2, 7/2)
%e | (3, 6) | (3, 3)
%e | (1, 2, 6) | (1, 1, 2)
%e | (1, 3, 5) | (1, 3/2, 5/3) (End)
%t Series[Sum[q^((n+1)(n+2)/2)/Product[1-q^(2k+1), {k, 0, n}], {n, 0, 12}], {q, 0, 100}]
%t nmax = 100; CoefficientList[Series[Sum[x^((k+1)*(k+2)/2) / Product[1-x^(2*j+1), {j, 0, k}], {k, 0, Floor[Sqrt[2*nmax]]}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 11 2019 *)
%Y Other '10th-order' mock theta functions are at A053281, A053283, A053284.
%K nonn,easy
%O 0,4
%A _Dean Hickerson_, Dec 19 1999
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