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%I M0685 #31 Dec 18 2021 23:39:43
%S 0,1,2,3,5,8,11,16,23,31,43,58,76,101,132,170,219,280,354,447,562,699,
%T 869,1076,1323,1625,1987,2418,2937,3556,4289,5162,6196,7413,8853,
%U 10547,12530,14860,17586,20763,24474,28792,33802,39624,46368,54163
%N Coefficients of the '2nd-order' mock theta function A(q).
%C The "second-order" mock theta function A(q). - _Jeremy Lovejoy_, Dec 19 2008
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 8.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vaclav Kotesovec, <a href="/A006304/b006304.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H G. E. Andrews, <a href="http://dx.doi.org/10.1007/BFb0096452">Mordell integrals and Ramanujan's "Lost" Notebook</a>, pp. 10-48 of Analytic Number Theory (Philadelphia 1980), Lect. Notes Math. 899 (1981).
%H R. J. McIntosh, <a href="http://dx.doi.org/10.4153/CMB-2007-028-9">Second order mock theta functions</a>, Canad. Math. Bull. 50 (2007), 284-290. [From _Jeremy Lovejoy_, Dec 19 2008]
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Mock_modular_form">Mock modular form</a>
%F 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))).
%F 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.
%F a(n) ~ exp(Pi*sqrt(n/2)) / (8*sqrt(n)). - _Vaclav Kotesovec_, Jun 11 2019
%e G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 11*x^6 + 16*x^7 + 23*x^8 + ...
%t 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}]
%t 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 *)
%t 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 *)
%Y Cf. A006305, A006306.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E Corrected and extended by _Dean Hickerson_, Dec 13 1999