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.

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

Offset: 0

Author

Michael Somos, Sep 19 2005

Keywords

• sign
• changed

Examples

			G.f. = 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 + ...
		

References

• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 53

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• S. Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328.

Crossrefs

Cf. A078905, A112274, A124242, A285348.

Programs

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))};

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.

a(n) = A112274(n) unless n=0.