|
|
A112803
|
|
Expansion of 1 + k(q) = 1 + r(q) * r(q^2)^2 in powers of q where r() is the Rogers-Ramanujan continued fraction.
|
|
5
|
|
|
1, 1, -1, -1, 2, 0, -2, 2, 1, -4, 1, 4, -4, -1, 6, -3, -6, 7, 3, -10, 4, 10, -12, -6, 18, -5, -18, 20, 8, -30, 10, 29, -31, -12, 46, -17, -44, 47, 20, -68, 23, 66, -72, -31, 104, -33, -98, 107, 44, -156, 51, 144, -154, -61, 220, -75, -206, 220, 90, -310, 104, 290, -312, -131, 442, -143, -408, 437, 178, -618, 202
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
REFERENCES
|
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 53
|
|
LINKS
|
|
|
FORMULA
|
Euler transform of period 10 sequence [1, -2, 0, 2, -2, 2, 0, -2, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2 - v)^2 - u*(2 - u*v).
Given g.f. k=A(x) then (k-1) * ((2-k) / k)^2 = B(x), (k-1)^2 * (k / (2-k)) = B(x^2) where B(x) = g.f. A078905.
G.f.: Product_{k>0} ((1 - x^(10*k - 2)) * (1 - x^(10*k - 5)) * (1 - x^(10*k - 8))^2) / ((1 - x^(10*k - 1)) * (1 - x^(10*k - 4))^2 * (1 - x^(10*k - 6))^2 * (1 - x^(10*k - 9))).
G.f.: (f(-x^5, -x^5) * f(-x^2, -x^8)^2) / (f(-x, -x^9) * f(-x^4, -x^6)^2) where f(,) is Ramanujan's two-variable theta function.
|
|
EXAMPLE
|
1 + x - x^2 - x^3 + 2*x^4 - 2*x^6 + 2*x^7 + x^8 - 4*x^9 + x^10 + 4*x^11 + ...
|
|
PROG
|
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( prod( k=1, n, (1 - x^k + A)^[0, -1, 2, 0, -2, 2, -2, 0, 2, -1][k%10 + 1]), n))}
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|