OFFSET
0,11
COMMENTS
REFERENCES
G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part I, Springer, 2005, see p. 57.
B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 9.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
O. Marichev and M. Trott, After 100 Years, Ramanujan Gap Filled
FORMULA
Given g.f. A(x), then B(x) = x * A(x^5) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = (u^3 - v) * (1 + u*v^3) - 3 * u^2*v^2.
Given g.f. A(x), then B(x) = x * A(x^5) satisfies 0 = f(B(x), B(x^5)) where f(u, v) = u^5 * (1 - 3*v + 4*v^2 - 2*v^3 + v^4) - v * (1 + 2*v + 4*v^2 + 3*v^3 + v^4).
Euler transform of period 20 sequence [1, 0, -1, -1, 0, 0, -1, 1, 1, 0, 1, 1, -1, 0, 0, -1, -1, 0, 1, 0, ...].
G.f.: (Sum_{k in Z} (-1)^k * (-x)^((5*k + 3)*k / 2)) / (Sum_{k in Z} (-1)^k * (-x)^((5*k + 1)*k / 2)).
G.f.: 1 / (1 - x / (1 + x^2 / (1 - x^3 / ...))). [continued fraction]
a(n) = (-1)^n * A007325(n).
G.f.: 1/G(0), where G(k)= 1 + (-x)^(k+1)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Jun 29 2013
EXAMPLE
G.f. = 1 - x + x^2 - x^4 + x^5 - x^6 + x^7 - x^9 + 2*x^10 - 3*x^11 + 2*x^12 + ...
G.f. = q - q^6 + q^11 - q^21 + q^26 - q^31 + q^36 - q^46 + 2*q^51 - 3*q^56 + 2*q^61 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ ContinuedFractionK[ (-q)^k, 1, {k, 0, n}], {q, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -q, -q^5] QPochhammer[ q^4, -q^5] / (QPochhammer[ q^2, -q^5] QPochhammer[-q^3, -q^5]), {q, 0, n}];
PROG
(PARI) {a(n) = my(k); if( n<0, 0, k = (3 + sqrtint(9 + 40*n)) \ 10; polcoeff( sum( n=-k, k, (-1)^n * (-x)^( (5*n^2 + 3*n) / 2), x * O(x^n)) / sum( n=-k, k, (-1)^n * (-x)^( (5*n^2 + n) / 2), x * O(x^n)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jun 10 2013
STATUS
approved