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A006304 Coefficients of the '2nd order' mock theta function A(q).
(Formerly M0685)
4
0, 1, 2, 3, 5, 8, 11, 16, 23, 31, 43, 58, 76, 101, 132, 170, 219, 280, 354, 447, 562, 699, 869, 1076, 1323, 1625, 1987, 2418, 2937, 3556, 4289, 5162, 6196, 7413, 8853, 10547, 12530, 14860, 17586, 20763, 24474, 28792, 33802, 39624, 46368, 54163 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The "second order" mock theta function A(q). [From Jeremy Lovejoy, Dec 19 2008]

REFERENCES

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 8.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

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

G. E. Andrews, Mordell integrals and Ramanujan's "Lost" Notebook, pp. 10-48 of Analytic Number Theory (Philadelphia 1980), Lect. Notes Math. 899 (1981).

R. J. McIntosh, Second order mock theta functions, Canad. Math. Bull. 50 (2007), 284-290. [From Jeremy Lovejoy, Dec 19 2008]

Wikipedia, Mock modular form

FORMULA

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

G.f.: sum for n >= 0 of q^(n+1)^2 (1+q)(1+q^3)...(1+q^(2n-1))/((1-q)(1-q^3)...(1-q^(2n+1)))^2

EXAMPLE

G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 11*x^6 + 16*x^7 + 23*x^8 + ...

MATHEMATICA

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

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k + 1)^2 QPochhammer[ -x, x^2, k] / QPochhammer[ x, x^2, k + 1]^2, {k, 0, Sqrt[ n] - 1}], {x, 0, n}]]; (* Michael Somos, Apr 08 2015 *)

CROSSREFS

Cf. A006305, A006306.

Sequence in context: A175831 A070228 A173599 * A238591 A039847 A046938

Adjacent sequences:  A006301 A006302 A006303 * A006305 A006306 A006307

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Corrected and extended by Dean Hickerson, Dec 13 1999

STATUS

approved

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Last modified December 8 02:07 EST 2016. Contains 278902 sequences.