A053254 Coefficients of the '3rd-order' mock theta function nu(q).

1, -1, 2, -2, 2, -3, 4, -4, 5, -6, 6, -8, 10, -10, 12, -14, 15, -18, 20, -22, 26, -29, 32, -36, 40, -44, 50, -56, 60, -68, 76, -82, 92, -101, 110, -122, 134, -146, 160, -176, 191, -210, 230, -248, 272, -296, 320, -350, 380, -410, 446, -484, 522, -566, 612, -660, 715, -772, 830, -896, 966, -1038

Offset

0,3

Comments

In Watson 1936 the function is denoted by upsilon(q). - Michael Somos, Jul 25 2015

References

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 31.

Links

G. C. Greubel, 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.

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

Formula

G.f.: nu(q) = Sum_{n >= 0} q^(n*(n+1))/((1+q)*(1+q^3)*...*(1+q^(2n+1)))

(-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.

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

a(2*n) = A085140(n). a(2*n + 1) = - A053253(n). - Michael Somos, Jul 25 2015

a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) / (2^(3/2)*sqrt(n)). - Vaclav Kotesovec, Jun 15 2019

Example

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 + ...

Mathematica

Series[Sum[q^(n(n+1))/Product[1+q^(2k+1), {k, 0, n}], {n, 0, 9}], {q, 0, 100}]

Prog

(PARI) /* Continued Fraction Expansion: */
{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

Crossrefs

Other '3rd-order' mock theta functions are at A000025, A053250, A053251, A053252, A053253, A053255.

Cf. A058140.

Sequence in context: A325358 A029146 A029053 * A067357 A051059 A132968

Adjacent sequences: A053251 A053252 A053253 * A053255 A053256 A053257

Keyword

sign,easy

Author

Dean Hickerson, Dec 19 1999

Status

approved