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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 [From Jeremy Lovejoy, Dec 19 2008] REFERENCES Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 8.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 Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe) 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 a(n) ~ exp(Pi*sqrt(n/2)) / (8*sqrt(n)). - Vaclav Kotesovec, Jun 11 2019 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 *) nmax = 100; CoefficientList[Series[Sum[x^(k+1)^2 * Product[1 + x^(2*j - 1), {j, 1, k}] / Product[1 - x^(2*j - 1), {j, 1, k+1}]^2, {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 11 2019 *) 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 EXTENSIONS Corrected and extended by Dean Hickerson, Dec 13 1999 STATUS approved

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Last modified September 19 04:52 EDT 2019. Contains 327187 sequences. (Running on oeis4.)