A279715 Expansion of a q-series used by Ramanujan in his Lost Notebook.

1, 2, 4, 6, 10, 16, 23, 34, 50, 70, 98, 136, 184, 250, 336, 444, 586, 768, 997, 1290, 1660, 2120, 2698, 3420, 4310, 5414, 6776, 8442, 10488, 12986, 16020, 19710, 24180, 29574, 36082, 43910, 53293, 64538, 77980, 94000, 113082, 135760, 162648, 194502, 232164

Offset: 0

Author

Michael Somos, Dec 17 2016

Keywords

• nonn

Examples

			G.f. = 1 + 2*x + 4*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 23*x^6 + 34*x^7 + ...
		

References

• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, page 1, 1st equation with a=1.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A256209.

Programs

Magma

m:=50; R:=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

Mathematica

a[ n_] := SeriesCoefficient[ Sum[ x^k QPochhammer[ -x, x, k] / QPochhammer[ x, x^2, k + 1] // FunctionExpand, {k, 0, n}], {x, 0, n}];

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))};

Formula

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))).