A053255 - OEIS

%I #22 Aug 12 2023 23:00:00

%S 1,-1,0,1,0,-1,1,-1,0,1,-1,0,2,-1,-1,1,-1,-1,2,-1,0,2,-1,-1,2,-2,-1,3,

%T -2,-1,3,-2,-1,3,-2,-1,4,-3,-1,4,-2,-2,4,-3,-2,5,-4,-2,6,-3,-2,6,-4,

%U -2,7,-5,-2,7,-5,-3,8,-6,-3,9,-6,-3,10,-6,-4,10,-7,-4,12,-8,-4,13,-8,-5,13,-9,-5,15,-10,-5,16,-11,-6,17,-12,-7,19,-13,-6,21,-13

%N Coefficients of the '3rd-order' mock theta function rho(q).

%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 15.

%H G. C. Greubel, <a href="/A053255/b053255.txt">Table of n, a(n) for n = 0..1000</a>

%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.: rho(q) = Sum_{n >= 0} q^(2*n*(n+1))/((1+q+q^2)*(1+q^3+q^6)*...*(1+q^(2*n+1)+q^(4*n+2))).

%t Series[Sum[q^(2n(n+1))/Product[1+q^(2k+1)+q^(4k+2), {k, 0, n}], {n, 0, 6}], {q, 0, 100}]

%Y Other '3rd-order' mock theta functions are at A000025, A053250, A053251, A053252, A053253, A053254.

%K sign,easy

%O 0,13

%A _Dean Hickerson_, Dec 19 1999