cp's OEIS Frontend

A285441 Expansion of q^(-2/5) * r(q)^2 * (1 + r(q) * r(q^2)^2) in powers of q where r() is the Rogers-Ramanujan continued fraction.

%I A285441 #27 Feb 16 2025 08:33:44
%S A285441 1,-1,0,2,-2,-2,5,-1,-6,7,2,-12,6,11,-15,-2,22,-14,-20,31,4,-41,24,37,
%T A285441 -58,-9,80,-44,-68,105,12,-143,83,119,-184,-16,238,-144,-196,307,30,
%U A285441 -391,234,317,-502,-49,638,-374,-511,804,68,-1014,600,802,-1254,-99,1562
%N A285441 Expansion of q^(-2/5) * r(q)^2 * (1 + r(q) * r(q^2)^2) in powers of q where r() is the Rogers-Ramanujan continued fraction.
%C A285441 G.f. A(q) satisfies: A(q) = q^(-2/5) * r(q)^2 * (1 + k(q)) = q^(-2/5) * r(q^2) * (1 - k(q)), where k(q) = r(q) * r(q^2)^2.
%H A285441 Seiichi Manyama, <a href="/A285441/b285441.txt">Table of n, a(n) for n = 0..5000</a>
%H A285441 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>
%o A285441 (Ruby)
%o A285441 def s(k, m, n)
%o A285441   s = 0
%o A285441   (1..n).each{|i| s += i if n % i == 0 && i % k == m}
%o A285441   s
%o A285441 end
%o A285441 def A007325(n)
%o A285441   ary = [1]
%o A285441   a = [0] + (1..n).map{|i| s(5, 1, i) + s(5, 4, i) - s(5, 2, i) - s(5, 3, i)}
%o A285441   (1..n).each{|i| ary << (1..i).inject(0){|s, j| s - a[j] * ary[-j]} / i}
%o A285441   ary
%o A285441 end
%o A285441 def mul(f_ary, b_ary, m)
%o A285441   s1, s2 = f_ary.size, b_ary.size
%o A285441   ary = Array.new(s1 + s2 - 1, 0)
%o A285441   (0..s1 - 1).each{|i|
%o A285441     (0..s2 - 1).each{|j|
%o A285441       ary[i + j] += f_ary[i] * b_ary[j]
%o A285441     }
%o A285441   }
%o A285441   ary[0..m]
%o A285441 end
%o A285441 def A285441(n)
%o A285441   ary1 = A007325(n)
%o A285441   ary2 = Array.new(n + 1, 0)
%o A285441   (0..n / 2).each{|i| ary2[i * 2] = ary1[i]}
%o A285441   ary = [-1] + mul(ary1, mul(ary2, ary2, n), n)[0..-2]
%o A285441   mul(ary2, (0..n).map{|i| -ary[i]}, n)
%o A285441 end
%o A285441 p A285441(100)
%Y A285441 Cf. A007325 (q^(-1/5) * r(q)), A055101, A112274 (k(q)), A112803 (1 + k(q)), A124242 (1 - k(q)), A285348, A285349.
%K A285441 sign
%O A285441 0,4
%A A285441 _Seiichi Manyama_, Apr 19 2017