

A053254


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


9



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
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OFFSET

0,3


COMMENTS

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


REFERENCES

Leila A. Dragonette, Some asymptotic formulae for the mock theta functions of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474500
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 31
George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 5580


LINKS

Table of n, a(n) for n=0..61.


FORMULA

G.f.: nu(q) = sum for n >= 0 of 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*(1x)/(1 + x^2*(1x^2)/(1 + x^3*(1x^3)/(1 + x^4*(1x^4)/(1 + x^5*(1x^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


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^(nk+1)*(1  x^(nk+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: A000929 A029146 A029053 * A067357 A051059 A132967
Adjacent sequences: A053251 A053252 A053253 * A053255 A053256 A053257


KEYWORD

sign,easy


AUTHOR

Dean Hickerson, Dec 19 1999


STATUS

approved



