cp's OEIS Frontend

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

%I A112274 #34 Aug 30 2025 17:40:01
%S A112274 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,
%T A112274 -18,20,8,-30,10,29,-31,-12,46,-17,-44,47,20,-68,23,66,-72,-31,104,
%U A112274 -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
%N A112274 Expansion of k(q) = r(q) * r(q^2)^2 in powers of q where r() is the Rogers-Ramanujan continued fraction.
%C A112274 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
%D A112274 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 53
%H A112274 Seiichi Manyama, <a href="/A112274/b112274.txt">Table of n, a(n) for n = 1..10000</a>
%H A112274 S. Cooper, <a href="https://dx.doi.org/10.1007/s11139-009-9198-5">On Ramanujan's function k(q)=r(q)r^2(q^2)</a>, Ramanujan J., 20 (2009), 311-328.
%H A112274 S. Cooper, <a href="https://www.researchgate.net/publication/266859001_Level_10_analogues_of_Ramanujan%27s_series_for_1p">Level 10 analogues of Ramanujan's series for 1/pi</a>, J. Ramanujan Math. Soc., 27 (2012), 75-92. See p. 77.
%F A112274 Euler transform of period 10 sequence [ -1, -1, 1, 1, 0, 1, 1, -1, -1, 0, ...].
%F A112274 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.
%F A112274 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + v)^2 - v * (1 - u^2).
%F A112274 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))).
%F A112274 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.
%F A112274 a(n) = A112803(n) unless n=0. - _Michael Somos_, Jul 08 2012
%F A112274 Convolution inverse is A214341. - _Michael Somos_, Jul 12 2012
%F A112274 k(q) = (r(q^2) - r(q)^2)/(r(q^2) + r(q)^2). - _Seiichi Manyama_, Apr 21 2017
%e A112274 G.f. = x - x^2 - x^3 + 2*x^4 - 2*x^6 + 2*x^7 + x^8 - 4*x^9 + x^10 + 4*x^11 + ...
%o A112274 (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))};
%Y A112274 Cf. A078905, A112803, A214341, A285554, A285555.
%K A112274 sign,changed
%O A112274 1,4
%A A112274 _Michael Somos_, Aug 30 2005