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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 (list; graph; refs; listen; history; text; internal format)
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 formulas for the mock theta functions of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474-500

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 31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) 55-80

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*(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

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: A000929 A029146 A029053 * A067357 A051059 A132968

Adjacent sequences:  A053251 A053252 A053253 * A053255 A053256 A053257

KEYWORD

sign,easy

AUTHOR

Dean Hickerson, Dec 19 1999

STATUS

approved

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Last modified December 9 12:25 EST 2016. Contains 278971 sequences.