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%I #36 Feb 02 2021 22:00:14
%S 1,1,-1,1,0,0,-1,1,0,1,-2,1,-1,2,-2,2,-1,1,-3,2,-1,3,-3,2,-2,3,-4,3,
%T -3,4,-5,5,-3,5,-7,5,-5,6,-7,7,-6,7,-9,9,-7,9,-11,9,-9,11,-13,12,-11,
%U 13,-15,15,-13,16,-19,17,-17,19,-21,21,-20,22,-26,25,-23,27,-30,29,-28,32,-35,34,-34,36,-41,40,-38,44,-48,46
%N Coefficients of the '5th-order' mock theta function f_0(q).
%C In Ramanujan's lost notebook page 21 is written the g.f. neatly crossed out between the 3rd and 4th equations. - _Michael Somos_, Feb 13 2017
%D Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22, 23.
%H Alois P. Heinz, <a href="/A053256/b053256.txt">Table of n, a(n) for n = 0..10000</a>
%H George E. Andrews, <a href="http://dx.doi.org/10.1090/S0002-9947-1986-0814916-2">The fifth and seventh order mock theta functions</a>, Trans. Amer. Math. Soc., 293 (1986) 113-134.
%H George E. Andrews and Frank G. Garvan, <a href="http://dx.doi.org/10.1016/0001-8708(89)90070-4">Ramanujan's "lost" notebook VI: The mock theta conjectures</a>, Advances in Mathematics, 73 (1989) 242-255.
%H Dean Hickerson, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00210590X">A proof of the mock theta conjectures</a>, Inventiones Mathematicae, 94 (1988) 639-660.
%H George N. Watson, <a href="https://doi.org/10.1112/plms/s2-42.1.274">The mock theta functions (2)</a>, Proc. London Math. Soc., series 2, 42 (1937) 274-304.
%F G.f.: 1 + Sum_{k>0} q^k^2 / ((1 + q) * (1 + q^2) * ... * (1 + q^k)).
%F Consider partitions of n into parts differing by at least 2. For n > 0: a(n) is the number of them with largest part odd minus number with largest part even.
%F a(n) ~ -(-1)^n * exp(Pi*sqrt(n/15)) / (2*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jun 15 2019
%e G.f. = 1 + x - x^2 + x^3 - x^6 + x^7 + x^9 - 2*x^10 + x^11 - x^12 + 2*x^13 - ...
%p N:= 100: # for a(0)..a(N)
%p g:= add(q^(k^2)/mul(1+q^i,i=1..k),k=0..floor(sqrt(N))):
%p S:= series(g,q,N+1):
%p seq(coeff(S,q,k),k=0..N)]; # _Robert Israel_, Mar 27 2018
%t Series[Sum[q^n^2/Product[1+q^k, {k, 1, n}], {n, 0, 10}], {q, 0, 100}]
%t a[ n_] := SeriesCoefficient[ Sum[ x^k^2 / QPochhammer[ -x, x, k] // FunctionExpand, {k, 0, Sqrt@ n}], {x, 0, n}]; (* _Michael Somos_, Feb 13 2017 *)
%o (PARI) {a(n) = my(t); if( n<0, 0, t = 1 + O(x^n); polcoeff( sum( k=1, sqrtint(n), t *= x^(2*k-1) / (1 + x^k + O(x^(n - (k-1)^2 + 1))), 1), n))}; /* _Michael Somos_, Mar 12 2006 */
%Y Other '5th-order' mock theta functions are at A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053266, A053267.
%K sign,easy
%O 0,11
%A _Dean Hickerson_, Dec 19 1999
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