This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A053254 #41 Jan 04 2025 06:12:31
%S A053254 1,-1,2,-2,2,-3,4,-4,5,-6,6,-8,10,-10,12,-14,15,-18,20,-22,26,-29,32,
%T A053254 -36,40,-44,50,-56,60,-68,76,-82,92,-101,110,-122,134,-146,160,-176,
%U A053254 191,-210,230,-248,272,-296,320,-350,380,-410,446,-484,522,-566,612,-660,715,-772,830,-896,966,-1038
%N A053254 Coefficients of the '3rd-order' mock theta function nu(q).
%C A053254 In Watson 1936 the function is denoted by upsilon(q). - _Michael Somos_, Jul 25 2015
%D A053254 George E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, (Example 6, p. 29).
%D A053254 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 31.
%H A053254 G. C. Greubel, <a href="/A053254/b053254.txt">Table of n, a(n) for n = 0..1000</a>
%H A053254 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 A053254 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 A053254 G.f.: nu(q) = Sum_{n >= 0} q^(n*(n+1))/((1+q)*(1+q^3)*...*(1+q^(2*n+1)))
%F A053254 (-1)^n*a(n) = number of partitions of n in which even parts are distinct and if k occurs then so does every positive even number less than k.
%F A053254 G.f.: 1/(1 + x*(1-x)/(1 + x^2*(1-x^2)/(1 + x^3*(1-x^3)/(1 + x^4*(1-x^4)/(1 + x^5*(1-x^5)/(1 + ...)))))), a continued fraction. - _Paul D. Hanna_, Jul 09 2013
%F A053254 a(2*n) = A085140(n). a(2*n + 1) = - A053253(n). - _Michael Somos_, Jul 25 2015
%F A053254 a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) / (2^(3/2)*sqrt(n)). - _Vaclav Kotesovec_, Jun 15 2019
%F A053254 From _Peter Bala_, Jan 03 2025: (Start)
%F A053254 a(n) = (-1)^n * A067357(n).
%F A053254 nu(-q) = Sum_{n >= 0} q^n * (1 + q)*(1 + q^3)*...*(1 + q^(2*n-1)) (Andrews, p. 29: in Example 6 take x = q and y = -q).
%F A053254 Conjecture: a(n) = (-1)^n * (A376628(n) + A376628(n+1)), or equivalently, (1 + q * nu(-q))/(1 + q) = Sum_{n >= 0} q^(n*(n+1))/((1 - q)*(1 - q^3)*...*(1 - q^(2*n-1))), the g.f. of A376628. (End)
%e A053254 G.f. = 1 - x + 2*x^2 - 2*x^3 + 2*x^4 - 3*x^5 + 4*x^6 - 4*x^7 + 5*x^8 + ...
%t A053254 Series[Sum[q^(n(n+1))/Product[1+q^(2k+1), {k, 0, n}], {n, 0, 9}], {q, 0, 100}]
%o A053254 (PARI) /* Continued Fraction Expansion: */
%o A053254 {a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 + x^(n-k+1)*(1 - x^(n-k+1))*CF+x*O(x^n))); polcoeff(CF, n)} \\ _Paul D. Hanna_, Jul 09 2013
%Y A053254 Other '3rd-order' mock theta functions are at A000025, A053250, A053251, A053252, A053253, A053255.
%Y A053254 Cf. A058140, A067357, A376628.
%K A053254 sign,easy
%O A053254 0,3
%A A053254 _Dean Hickerson_, Dec 19 1999