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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 - Jonathan Vos Post, Sep 01 2005 REFERENCES Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 53 LINKS Seiichi Manyama, Table of n, a(n) for n = 1..10000 S. Cooper, On Ramanujan's function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 * A181391 A333359 A262670 Adjacent sequences:  A112271 A112272 A112273 * A112275 A112276 A112277 KEYWORD sign AUTHOR Michael Somos, Aug 30 2005 STATUS approved

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Last modified August 14 13:16 EDT 2020. Contains 336480 sequences. (Running on oeis4.)