A292445 Expansion of a q-series used by Ramanujan in his Lost Notebook.

1, 4, 10, 22, 44, 82, 145, 248, 410, 658, 1036, 1598, 2420, 3614, 5322, 7738, 11132, 15850, 22353, 31260, 43366, 59708, 81650, 110932, 149788, 201112, 268562, 356790, 471732, 620834, 813480, 1061496, 1379626, 1786282, 2304440, 2962566, 3795921, 4848160

Offset: 0

Author

Michael Somos, Sep 16 2017

Keywords

• nonn

Comments

Similar to A292420 but with a=1.

Examples

			G.f. = 1 + 4*x + 10*x^2 + 22*x^3 + 44*x^4 + 82*x^5 + 145*x^6 + ...
		

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• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, page 1, 1st equation with a=1.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..2500

Crossrefs

Cf. A022567, A053253, A256209, A279715, A292420.

Programs

Mathematica

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

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^2 * sum(k=0, n, x^k / prod(i=0, k, 1 - x^(2*i+1), 1 + A/x^k)), n))};

Formula

a(n) = 2 * A256209(n) - A279715(n).

G.f. is the product of the g.f. of A022567 and A053253.