A053250 Coefficients of the '3rd-order' mock theta function phi(q).

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, 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, 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

Offset

0,10

References

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

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

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 17, 31.

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

Leila A. Dragonette, Some asymptotic formulas for the mock theta series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474-500.

John F. R. Duncan, Michael J. Griffin and Ken Ono, Proof of the Umbral Moonshine Conjecture, arXiv:1503.01472 [math.RT], 2015.

A. Folsom, K. Ono and R. C. Rhoades, Ramanujan's radial limits, 2013. - From N. J. A. Sloane, Feb 09 2013

George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.

Formula

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).

a(n) = (-1)^n*(A027358(n)-A027357(n)). - Vladeta Jovovic, Mar 12 2006

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

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

Example

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 + ...

Maple

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

Mathematica

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

Prog

(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 */

Crossrefs

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

Cf. A207569, A215066.

Sequence in context: A225152 A117447 A328775 * A364259 A302242 A336915

Adjacent sequences: A053247 A053248 A053249 * A053251 A053252 A053253

Keyword

sign,easy

Author

Dean Hickerson, Dec 19 1999

Status

approved