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A112274 Expansion of k(q) = r(q) * r(q^2)^2 in powers of q where r() is the Rogers-Ramanujan continued fraction. 12
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 (list; graph; refs; listen; history; text; internal format)
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 53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
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
Sequence in context: A321594 A112803 A124242 * A336891 A181391 A333359
KEYWORD
sign
AUTHOR
Michael Somos, Aug 30 2005
STATUS
approved

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)