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

0,2

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, page 1, 1st equation with a=1.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

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))).

Example

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

Mathematica

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

Prog

(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))};
(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

Crossrefs

Cf. A256209.

Sequence in context: A073150 A076529 A323283 * A132212 A327047 A332281

Adjacent sequences: A279712 A279713 A279714 * A279716 A279717 A279718

Keyword

nonn

Author

Michael Somos, Dec 17 2016

Status

approved