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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, page 1, 1st equation with a=-1.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
KEYWORD
sign
AUTHOR
Michael Somos, Sep 17 2017
STATUS
approved