login
Coefficients of the '6th-order' mock theta function 2 mu(q).
9

%I #27 Jan 31 2021 22:31:09

%S 1,2,-3,4,-4,6,-11,14,-15,22,-31,34,-41,56,-69,82,-98,120,-152,178,

%T -204,254,-308,354,-415,496,-587,680,-785,922,-1084,1248,-1427,1664,

%U -1935,2202,-2517,2906,-3336,3798,-4315,4930,-5636,6380,-7202,8194,-9305,10474,-11801,13342,-15050

%N Coefficients of the '6th-order' mock theta function 2 mu(q).

%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 13

%H Vaclav Kotesovec, <a href="/A053273/b053273.txt">Table of n, a(n) for n = 0..1000</a> (corrected previous b-file 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.: 2 mu(q) = 1 + Sum_{n >= 0} (-1)^n q^(n+1) (1+q^n) (1-q)(1-q^3)...(1-q^(2n-1))/((1+q)(1+q^2)...(1+q^(n+1))).

%F a(n) ~ -(-1)^n * exp(Pi*sqrt(n/3)) / (2*sqrt(3*n)). - _Vaclav Kotesovec_, Jun 15 2019

%t Series[1+Sum[(-1)^n q^(n+1) (1+q^n) Product[1-q^k, {k, 1, 2n-1, 2}]/Product[1+q^k, {k, 1, n+1}], {n, 0, 99}], {q, 0, 100}]

%t nmax = 100; CoefficientList[Series[1 + Sum[(-1)^k * x^(k+1) * (1+x^k) * Product[1-x^j, {j, 1, 2*k-1, 2}] / Product[1+x^j, {j, 1, k+1}], {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 15 2019 *)

%Y Other '6th-order' mock theta functions are at A053268, A053269, A053270, A053271, A053272, A053274.

%K sign,easy

%O 0,2

%A _Dean Hickerson_, Dec 19 1999