A124242 -id:A124242 - cp's OEIS frontend

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.

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, -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

Offset: 0

Michael Somos, Sep 19 2005

			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 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 53

Seiichi Manyama, Table of n, a(n) for n = 0..10000
S. Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328.

Cf. A078905, A112274, A124242, A285348.

{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))};

Euler transform of period 10 sequence [1, -2, 0, 2, -2, 2, 0, -2, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2 - v)^2 - u*(2 - u*v).
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.
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))).
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.
a(n) = A112274(n) unless n=0.

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.

Original entry on oeis.org

1, -1, 0, 2, -2, -2, 5, -1, -6, 7, 2, -12, 6, 11, -15, -2, 22, -14, -20, 31, 4, -41, 24, 37, -58, -9, 80, -44, -68, 105, 12, -143, 83, 119, -184, -16, 238, -144, -196, 307, 30, -391, 234, 317, -502, -49, 638, -374, -511, 804, 68, -1014, 600, 802, -1254, -99, 1562

Offset: 0

Seiichi Manyama, Apr 19 2017

sign

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.

Seiichi Manyama, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

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

def s(k, m, n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0 && i % k == m}
  s
end
def A007325(n)
  ary = [1]
  a = [0] + (1..n).map{|i| s(5, 1, i) + s(5, 4, i) - s(5, 2, i) - s(5, 3, i)}
  (1..n).each{|i| ary << (1..i).inject(0){|s, j| s - a[j] * ary[-j]} / i}
  ary
end
def mul(f_ary, b_ary, m)
  s1, s2 = f_ary.size, b_ary.size
  ary = Array.new(s1 + s2 - 1, 0)
  (0..s1 - 1).each{|i|
    (0..s2 - 1).each{|j|
      ary[i + j] += f_ary[i] * b_ary[j]
    }
  }
  ary[0..m]
end
def A285441(n)
  ary1 = A007325(n)
  ary2 = Array.new(n + 1, 0)
  (0..n / 2).each{|i| ary2[i * 2] = ary1[i]}
  ary = [-1] + mul(ary1, mul(ary2, ary2, n), n)[0..-2]
  mul(ary2, (0..n).map{|i| -ary[i]}, n)
end
p A285441(100)

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

PARI

Formula