A292511 Expansion of a q-series used by Ramanujan in his Lost Notebook.

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, -8, 6, -10, 6, -5, 15, -8, 9, -10, 7, -7, 12, -10, 3, -11, 15, -7, 15, -8, 6, -13, 12, -9, 12, -9, 9, -14, 12, -7, 15, -12, 6, -15, 13, -6, 21, -12, 12, -13, 6, -11

Offset

0,3

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, page 1, 1st equation with a=-1.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

Formula

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

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) + ...

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

Example

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 + ...

Mathematica

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}];
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 *)

Prog

(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))};
(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 */

Crossrefs

Cf. A015128, A260195, A292526.

Sequence in context: A144733 A091460 A035093 * A238872 A260195 A061266

Adjacent sequences: A292508 A292509 A292510 * A292512 A292513 A292514

Keyword

sign

Author

Michael Somos, Sep 17 2017

Status

approved