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%I #10 Sep 08 2022 08:46:18
%S 1,2,4,6,10,16,23,34,50,70,98,136,184,250,336,444,586,768,997,1290,
%T 1660,2120,2698,3420,4310,5414,6776,8442,10488,12986,16020,19710,
%U 24180,29574,36082,43910,53293,64538,77980,94000,113082,135760,162648,194502,232164
%N Expansion of a q-series used by Ramanujan in his Lost Notebook.
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, page 1, 1st equation with a=1.
%H G. C. Greubel, <a href="/A279715/b279715.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: Sum_{k>=0} x^k * (1 + x) * (1 + x^2) * ... * (1 + x^k) / ((1 - x) * (1 - x^3) * ... * (1 - x^(2*k+1))).
%e G.f. = 1 + 2*x + 4*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 23*x^6 + 34*x^7 + ...
%t a[ n_] := SeriesCoefficient[ Sum[ x^k QPochhammer[ -x, x, k] / QPochhammer[ x, x^2, k + 1] // FunctionExpand, {k, 0, n}], {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum(k=0, n, x^k * prod(i=1, k, 1 + x^i, 1 + A) / prod(i=1, k+1, 1 - x^(2*i-1), 1 + A), A), n))};
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[x^k* (&*[(1+x^j): j in [0..k]])/(&*[(1-x^(2*s+1)): s in [0..k]]) : k in [0..(m+2)]])/2 )); // _G. C. Greubel_, Nov 12 2018
%Y Cf. A256209.
%K nonn
%O 0,2
%A _Michael Somos_, Dec 17 2016
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