A053271 - OEIS

%I #23 Jun 12 2019 09:01:52

%S 0,1,1,2,3,3,5,7,8,11,14,17,22,28,33,41,51,60,74,89,105,127,151,177,

%T 210,248,289,340,398,461,537,624,719,832,960,1101,1267,1453,1660,1899,

%U 2167,2465,2807,3190,3614,4097,4638,5237,5915,6671,7507,8450,9498

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

%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 13.

%H Vaclav Kotesovec, <a href="/A053271/b053271.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.: sigma(q) = Sum_{n >= 0}  q^((n+1)(n+2)/2) (1+q)(1+q^2)...(1+q^n)/((1-q)(1-q^3)...(1-q^(2n+1))).

%F a(n) ~ exp(Pi*sqrt(n/3)) / (4*sqrt(3*n)). - _Vaclav Kotesovec_, Jun 12 2019

%t Series[Sum[q^((n+1)(n+2)/2) Product[1+q^k, {k, 1, n}]/Product[1-q^k, {k, 1, 2n+1, 2}], {n, 0, 12}], {q, 0, 100}]

%t nmax = 100; CoefficientList[Series[Sum[x^((k+1)*(k+2)/2) * Product[1+x^j, {j, 1, k}]/Product[1-x^j, {j, 1, 2*k+1, 2}], {k, 0, Floor[Sqrt[2*nmax]]}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 12 2019 *)

%Y Other '6th-order' mock theta functions are at A053268, A053269, A053270, A053272, A053273, A053274.

%K nonn,easy

%O 0,4

%A _Dean Hickerson_, Dec 19 1999