A285441 - OEIS

%I #26 Oct 24 2018 02:32:50

%S 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 -58,-9,80,-44,-68,105,12,-143,83,119,-184,-16,238,-144,-196,307,30,

%U -391,234,317,-502,-49,638,-374,-511,804,68,-1014,600,802,-1254,-99,1562

%N 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 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 Seiichi Manyama, <a href="/A285441/b285441.txt">Table of n, a(n) for n = 0..5000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>

%o (Ruby)

%o def s(k, m, n)

%o   s = 0

%o   (1..n).each{|i| s += i if n % i == 0 && i % k == m}

%o   s

%o end

%o def A007325(n)

%o   ary = [1]

%o   a = [0] + (1..n).map{|i| s(5, 1, i) + s(5, 4, i) - s(5, 2, i) - s(5, 3, i)}

%o   (1..n).each{|i| ary << (1..i).inject(0){|s, j| s - a[j] * ary[-j]} / i}

%o   ary

%o end

%o def mul(f_ary, b_ary, m)

%o   s1, s2 = f_ary.size, b_ary.size

%o   ary = Array.new(s1 + s2 - 1, 0)

%o   (0..s1 - 1).each{|i|

%o     (0..s2 - 1).each{|j|

%o       ary[i + j] += f_ary[i] * b_ary[j]

%o     }

%o   }

%o   ary[0..m]

%o end

%o def A285441(n)

%o   ary1 = A007325(n)

%o   ary2 = Array.new(n + 1, 0)

%o   (0..n / 2).each{|i| ary2[i * 2] = ary1[i]}

%o   ary = [-1] + mul(ary1, mul(ary2, ary2, n), n)[0..-2]

%o   mul(ary2, (0..n).map{|i| -ary[i]}, n)

%o end

%o p A285441(100)

%Y Cf. A007325 (q^(-1/5) * r(q)), A055101, A112274 (k(q)), A112803 (1 + k(q)), A124242 (1 - k(q)), A285348, A285349.

%K sign

%O 0,4

%A _Seiichi Manyama_, Apr 19 2017