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%I #27 Jan 31 2021 20:51:48
%S 0,1,-1,1,-2,3,-2,2,-4,5,-5,5,-7,9,-8,9,-12,14,-15,16,-20,23,-23,25,
%T -31,36,-37,40,-47,54,-56,60,-71,79,-84,91,-103,115,-121,131,-149,164,
%U -174,188,-210,232,-245,264,-294,321,-343,368,-406,443,-470,505,-554,602,-641,687,-751,813,-863,925
%N Coefficients of the '6th-order' mock theta function psi(q).
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 4, 13
%H Vaclav Kotesovec, <a href="/A053269/b053269.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)
%H George E. Andrews and Dean Hickerson, <a href="https://doi.org/10.1016/0001-8708(91)90083-J">Ramanujan's "lost" notebook VII: The sixth order mock theta functions</a>, Advances in Mathematics, 89 (1991) 60-105.
%F G.f.: psi(q) = Sum_{n >= 0} (-1)^n q^(n+1)^2 (1-q)*(1-q^3)...(1-q^(2n-1)) /((1+q)*(1+q^2)...(1+q^(2n+1))).
%F a(3*n + 1) = A262614(n). a(3*n + 2) = - A263041(n). - _Michael Somos_, Apr 17 2016
%F a(n) ~ -(-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(3*n)). - _Vaclav Kotesovec_, Jun 15 2019
%t Series[Sum[(-1)^n q^(n+1)^2 Product[1-q^k, {k, 1, 2n-1, 2}]/Product[1+ q^k, {k, 1, 2n+1}], {n, 0, 9}], {q, 0, 100}]
%t nmax = 100; CoefficientList[Series[Sum[(-1)^k * x^((k+1)^2) * Product[1-x^j, {j, 1, 2*k-1, 2}]/Product[1+ x^j, {j, 1, 2*k+1}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 15 2019 *)
%Y Other '6th-order' mock theta functions are at A053268, A053270, A053271, A053272, A053273, A053274.
%Y Cf. A262614, A263041.
%K sign,easy
%O 0,5
%A _Dean Hickerson_, Dec 19 1999
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