A292511 - OEIS

%I #16 Jul 29 2018 20:46:37

%S 1,-1,3,-2,3,-3,4,-3,6,-4,3,-5,6,-4,9,-5,3,-7,7,-5,9,-6,6,-8,9,-5,9,

%T -8,6,-10,6,-5,15,-8,9,-10,7,-7,12,-10,3,-11,15,-7,15,-8,6,-13,12,-9,

%U 12,-9,9,-14,12,-7,15,-12,6,-15,13,-6,21,-12,12,-13,6,-11

%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%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, page 1, 1st equation with a=-1.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="/A292511/b292511.txt">Table of n, a(n) for n = 0..2500</a>

%F a(n-1) = -(-1)^n * A260195(n) for all n in Z.

%F G.f.: 1 / (1+x)^2 + x * (1+x^2) / ((1+x) * (1+x^3)^2) + x^2 * (1+x^2) * (1+x^4) / ((1+x) * (1+x^3) * (1+x^5)^2) + ...

%F G.f. is the product of the g.f. of A015128 and the g.f. of A292526 divided by x. - _Michael Somos_, Sep 18 2017

%e G.f. = 1 - x + 3*x^2 - 2*x^3 + 3*x^4 - 3*x^5 + 4*x^6 - 3*x^7 + 6*x^8 - 4*x^9 + ...

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

%t a[ n_] := If[ n < 1, Boole[n==0], SeriesCoefficient[ QPochhammer[ x^2] / QPochhammer[ x]^2 Sum[ -(-1)^k x^k^2 (1 - x^(2 k - 1)) / (1 + x^(2 k - 1))^2, {k, Sqrt@n}] / x, {x, 0, n}]]; (* _Michael Somos_, Sep 18 2017 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, n, x^k * prod(i=1, min(k, n-k), (1 + x^(2*i)) / (1 + x^(2*i-1)), 1 / (1 + x^(2*k+1) + x*O(x^(n-k)))^2)), n))};

%o (PARI) {a(n) = my(A); if( n<0, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A)^2 * sum(k=1, sqrtint(n), -(-1)^k * x^(k^2) * (1 - x^(2*k-1) + x * O(x^(n-k^2))) / (1 + x^(2*k-1))^2), n))}; /* _Michael Somos_, Sep 18 2017 */

%Y Cf. A015128, A260195, A292526.

%K sign

%O 0,3

%A _Michael Somos_, Sep 17 2017