A112274 Expansion of k(q) = r(q) * r(q^2)^2 in powers of q where r() is the Rogers-Ramanujan continued fraction.

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, 567

Offset

1,4

Comments

Cumulative sums are: 1, 0, -1, 1, 1, -1, 1, 2, -2, -1, 3, -1, -2, 4, 1, -5, ...-5, 2, 5, -5, -1, 9, -3, -9, 9, 4, -14, 6, 14, -16, -6, 23. Conjecture: limit_[n goes to infinity] (cumulative sum of A112274)/n = 0. - Jonathan Vos Post, Sep 01 2005

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 53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 = 1..10000

S. Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328.

S. Cooper, Level 10 analogues of Ramanujan's series for 1/pi, J. Ramanujan Math. Soc., 27 (2012), 75-92. See p. 77.

Formula

Euler transform of period 10 sequence [ -1, -1, 1, 1, 0, 1, 1, -1, -1, 0, ...].

Expansion of x * (f(-x^2, -x^8) * f(-x, -x^9)) / (f(-x^4, -x^6) * f(-x^3, -x^7)) in powers of x where f(,) is Ramanujan's two-variable theta function.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + v)^2 - v * (1 - u^2).

G.f.: x * Product_{k>0} (1 - x^(10*k - 1)) * (1 - x^(10*k - 2)) * (1 - x^(10*k - 8)) * (1 - x^(10*k - 9)) / ((1 - x^(10*k - 3)) * (1 - x^(10*k - 4)) * (1 - x^(10*k - 6)) * (1 - x^(10*k - 7))).

Given g.f. k = A(x) then k * ((1 - k) / (1 + k))^2 = B(x), k^2 * ((1 + k) / (1 - k)) = B(x^2) where B(x) = g.f. A078905.

a(n) = A112803(n) unless n=0. - Michael Somos, Jul 08 2012

Convolution inverse is A214341. - Michael Somos, Jul 12 2012

k(q) = (r(q^2) - r(q)^2)/(r(q^2) + r(q)^2). - Seiichi Manyama, Apr 21 2017

Example

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<1, 0, n--; A = x * O(x^n); polcoeff( prod( k=1, n, (1 - x^k + A)^[0, 1, 1, -1, -1, 0, -1, -1, 1, 1][k%10 + 1]), n))}

Crossrefs

Cf. A078905, A112803, A214341, A285554, A285555.

Sequence in context: A321594 A112803 A124242 * A336891 A181391 A333359

Adjacent sequences: A112271 A112272 A112273 * A112275 A112276 A112277

Keyword

sign

Author

Michael Somos, Aug 30 2005

Status

approved