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A006304
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Coefficients of the '2nd-order' mock theta function A(q).
(Formerly M0685)
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4
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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
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OFFSET
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0,3
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COMMENTS
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The "second-order" mock theta function A(q). - Jeremy Lovejoy, Dec 19 2008
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REFERENCES
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 8.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).
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LINKS
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FORMULA
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G.f.: Sum_{n>=0} q^(n+1) (1+q^2)(1+q^4)...(1+q^(2n))/((1-q)(1-q^3)...(1-q^(2n+1))).
G.f.: Sum_{n>=0} q^(n+1)^2 (1+q)(1+q^3)...(1+q^(2n-1))/((1-q)(1-q^3)...(1-q^(2n+1)))^2.
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EXAMPLE
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G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 11*x^6 + 16*x^7 + 23*x^8 + ...
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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