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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.

%I A112803 #25 Aug 30 2025 17:13:51
%S A112803 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,
%T A112803 -18,20,8,-30,10,29,-31,-12,46,-17,-44,47,20,-68,23,66,-72,-31,104,
%U A112803 -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
%N 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.
%D A112803 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 53
%H A112803 Seiichi Manyama, <a href="/A112803/b112803.txt">Table of n, a(n) for n = 0..10000</a>
%H A112803 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.
%F A112803 Euler transform of period 10 sequence [1, -2, 0, 2, -2, 2, 0, -2, 1, 0, ...].
%F A112803 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2 - v)^2 - u*(2 - u*v).
%F A112803 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.
%F A112803 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))).
%F A112803 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.
%F A112803 a(n) = A112274(n) unless n=0.
%e A112803 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 + ...
%o A112803 (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))};
%Y A112803 Cf. A078905, A112274, A124242, A285348.
%K A112803 sign,changed
%O A112803 0,5
%A A112803 _Michael Somos_, Sep 19 2005