%I #28 Feb 02 2021 20:31:43
%S 0,1,1,1,1,2,2,2,2,3,3,4,4,5,5,6,6,8,8,9,10,12,12,14,15,17,18,20,21,
%T 25,26,29,31,35,36,41,43,48,51,56,59,66,70,76,81,89,94,103,109,119,
%U 126,137,144,158,167,180,191,207,218,236,250,269,285,306,323,349,368
%N Coefficients of the '5th-order' mock theta function Phi(q).
%C In Ramanujan's lost notebook the generating function is denoted by phi(q) on pages 18 and 20, however on page 18 there is no minus one first term. - _Michael Somos_, Jul 07 2015
%D G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, MR2952081, See p. 12, Equation (2.1.18) and also page 26 equation (2.4.8).
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 18, 20, 23.
%H Vaclav Kotesovec, <a href="/A053266/b053266.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)
%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.
%F G.f.: -1 + Sum_{k>=0} q^(5k^2)/((1-q)(1-q^4)(1-q^6)(1-q^9)...(1-q^(5k+1))).
%F 3*a(n) = A053262(n) + A259910(n) unless n=0. - _Michael Somos_, Jul 07 2015
%F a(n) ~ sqrt(phi/2) * exp(Pi*sqrt(2*n/15)) / (5^(3/4) * sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jun 12 2019
%e G.f. = x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 3*x^9 + ...
%t a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[1 + 24 n/5]}, SeriesCoefficient[ -1 + Sum[ (-1)^k x^(5 k (3 k + 1)/2) / (1 - x^(5 k + 1)), {k, Quotient[m + 1, -6], Quotient[m - 1, 6]}] / QPochhammer[ x^5], {x, 0, n}]]]; (* _Michael Somos_, Jul 07 2015 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n\5), x^(5*k^2) / prod(i=1, 5*k+1, 1 - if( i%5==1 || i%5==4, x^i), 1 + x * O(x^(n - 5*k^2)))) - 1, n))}; /* _Michael Somos_, Jul 07 2015 */
%o (PARI) {a(n) = my(A, m); if( n<0, 0, m = sqrtint(1 + 24*n\5); A = x * O(x^n); polcoeff( sum(k=(m + 1)\-6, (m - 1)\6, (-1)^k * x^(5*k*(3*k + 1)/2) / (1 - x^(5*k + 1)), A) / eta(x^5 + A) - 1, n))}; /* _Michael Somos_, Jul 07 2015 */
%Y Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053267.
%Y Cf. A259910.
%K nonn,easy
%O 0,6
%A _Dean Hickerson_, Dec 19 1999