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%I #21 Jun 12 2019 10:33:25
%S 1,1,1,1,1,2,2,2,2,3,3,4,4,5,5,6,6,8,8,9,10,12,12,14,15,17,18,20,21,
%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
%N Coefficients of the '5th-order' mock theta function Phi(q) with a(0)=1.
%C In Ramanujan's lost notebook the generating function is denoted by phi(q) on pages 18 and 20, however on page 20 there is a minus one first term.
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 18, 20, 23
%H G. C. Greubel, <a href="/A279135/b279135.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: Sum_{k>=0} x^(5*k^2) / ((1 - x) * (1 - x^4) * (1 - x^6) * (1 - x^9)...(1 - x^(5*k+1))).
%F 3*a(n) = A053262(n) + A259910(n) unless n=0. [Ramanujan, p. 23, equation 6]
%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. = 1 + 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, SeriesCoefficient[ Sum[ x^(5 k^2) / (QPochhammer[ x, x^5, k + 1] QPochhammer[ x^4, x^5, k]) // FunctionExpand, {k, 0, Sqrt[n/5]}], {x, 0, n}]];
%t a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[1 + 24 n/5]}, SeriesCoefficient[ 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}]]];
%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)))), n))};
%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), n))};
%Y Cf. A053262. Essentially the same as A053266.
%Y Cf. A259910.
%K nonn
%O 0,6
%A _Michael Somos_, Dec 06 2016
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