A053250 - OEIS

%I #40 Aug 12 2023 23:00:17

%S 1,1,0,-1,1,1,-1,-1,0,2,0,-2,1,1,-1,-2,1,3,-1,-2,1,2,-2,-3,1,4,0,-4,2,

%T 3,-2,-4,1,5,-2,-5,3,5,-3,-5,2,7,-2,-7,3,6,-4,-8,3,9,-2,-9,5,9,-5,-10,

%U 3,12,-4,-12,5,11,-6,-13,6,16,-6,-15,7,15,-8,-17,7,19,-6,-20,9,19,-10,-22,8,25,-9,-25,12,25,-12,-27,11,31

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

%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.12), p. 58, Eq. (26.56).

%D Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355

%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 17, 31.

%H T. D. Noe, <a href="/A053250/b053250.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 A. Folsom, K. Ono and R. C. Rhoades, <a href="http://math.stanford.edu/~rhoades/FILES/ramanujans_radial.pdf">Ramanujan's radial limits, 2013</a>. - From _N. J. A. Sloane_, Feb 09 2013

%H George N. Watson, <a href="http://jlms.oxfordjournals.org/content/s1-11/1/55.extract">The final problem: an account of the mock theta functions</a>, J. London Math. Soc., 11 (1936) 55-80.

%F Consider partitions of n into distinct odd parts. a(n) = number of them for which the largest part minus twice the number of parts is == 3 (mod 4) minus the number for which it is == 1 (mod 4).

%F a(n) = (-1)^n*(A027358(n)-A027357(n)). - _Vladeta Jovovic_, Mar 12 2006

%F G.f.: 1 + Sum_{k>0} x^k^2 / ((1 + x^2) (1 + x^4) ... (1 + x^(2*k))).

%F G.f. 1 + Sum_{n >= 0} x^(2*n+1)*Product_{k = 1..n} (x^(2*k-1) - 1) (Folsom et al.). Cf. A207569 and A215066. - _Peter Bala_, May 16 2017

%e G.f. = 1 + x - x^3 + x^4 + x^5 - x^6 - x^7 + 2*x^9 - 2*x^11 + x^12 + x^13 - x^14 + ...

%p f:=n->q^(n^2)/mul((1+q^(2*i)),i=1..n); add(f(n),n=0..10);

%t Series[Sum[q^n^2/Product[1+q^(2k), {k, 1, n}], {n, 0, 10}], {q, 0, 100}]

%t a[ n_] := SeriesCoefficient[ Sum[ x^k^2 / QPochhammer[ -x^2, x^2, k], {k, 0, Sqrt@ n}], {x, 0, n}]; (* _Michael Somos_, Jul 09 2015 *)

%o (PARI) {a(n) = my(t); if(n<0, 0, t = 1 + O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= x^(2*k - 1) / (1 + x^(2*k)) + O(x^(n - (k-1)^2 + 1)), 1), n))}; /* _Michael Somos_, Jul 16 2007 */

%Y Other '3rd-order' mock theta functions are at A000025, A053251, A053252, A053253, A053254, A053255.

%Y Cf. A207569, A215066.

%K sign,easy

%O 0,10

%A _Dean Hickerson_, Dec 19 1999