A112803 -id:A112803 - 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.

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, 567

Offset: 1

Michael Somos, Aug 30 2005

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

			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 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 53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 = 1..10000
S. Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328.
S. Cooper, Level 10 analogues of Ramanujan's series for 1/pi, J. Ramanujan Math. Soc., 27 (2012), 75-92. See p. 77.

Cf. A078905, A112803, A214341, A285554, A285555.

{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))};

Euler transform of period 10 sequence [ -1, -1, 1, 1, 0, 1, 1, -1, -1, 0, ...].
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.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + v)^2 - v * (1 - u^2).
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))).
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.
a(n) = A112803(n) unless n=0. - Michael Somos, Jul 08 2012
Convolution inverse is A214341. - Michael Somos, Jul 12 2012
k(q) = (r(q^2) - r(q)^2)/(r(q^2) + r(q)^2). - Seiichi Manyama, Apr 21 2017

A285348 Expansion of r(q^2) / r(q)^2 in powers of q where r() is the Rogers-Ramanujan continued fraction.

Original entry on oeis.org

1, 2, 0, -4, -2, 6, 8, -4, -16, -6, 20, 24, -12, -44, -16, 52, 62, -28, -108, -40, 122, 144, -64, -244, -88, 266, 308, -136, -508, -180, 544, 624, -272, -1008, -356, 1060, 1206, -524, -1920, -672, 1988, 2244, -968, -3524, -1224, 3606, 4048, -1732, -6284

Offset: 0

Seiichi Manyama, Apr 17 2017

sign

Let k(q) = r(q) * r(q^2)^2.
G.f. satisfies: A(q) = (1 + k(q))/(1 - k(q)).
And r(q^2)^5 = k(q)^2 * A(q).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Wikipedia, Rogers-Ramanujan continued fraction

r(q^k) / r(q)^k: this sequence (k=2), A285583 (k=3), A285584 (k=4), A285585 (k=5).

Cf. A007325, A078905 (r(q)^5), A112274 (k(q)), A112803 (1 + k(q)), A285349, A285355 (k(q)^2).

a(n) = A285349(n) - A138518(n) for n>0 (conjectured). - Thomas Baruchel, May 14 2018

A124242 Expansion of a parametrization of Ramanujan's continued fraction.

Original entry on oeis.org

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, Oct 27 2006

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			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
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A078905, A112274, A112803, A285348.

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 1, -1, -2, 1, 2, 1, -2, -1, 1, 0}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, Jan 06 2016 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoef( prod(k=1, n, (1 - x^k + A)^[0, 1, -1, -2, 1, 2, 1, -2, -1, 1][k%10+1]), n))};

Euler transform of period 10 sequence [ -1, 1, 2, -1, -2, -1, 2, 1, -1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v^2 - (2-u) * (2 - (2-u) * (2-v)).
Given g.f. A(x) =: k, then B(x) = (1-k) * (k / (2-k))^2, B(x^2) = (1-k)^2 * ((2-k) / k) where B(x) is the g.f. for A078905.
Expansion of f(-x^5, -x^10)^3 / (f(x, x^4) * f(-x^3, -x^7)^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Jan 06 2016
G.f.: Product_{k>0} ((1 - x^(10*k-5)) / ((1 - x^(10*k-3)) * (1 - x^(10*k-7))))^2 * (1 - x^(10*k-1)) * (1 - x^(10*k-4)) * (1 - x^(10*k-6)) * (1 - x^(10*k-9)) / ((1-x^(10*k-2)) * (1-x^(10*k-8))).
-a(n) = A112274(n) unless n = 0.
G.f.: 1 - r(q) * r(q^2)^2 where r() is the Rogers-Ramanujan continued fraction. - Seiichi Manyama, Apr 18 2017

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)

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

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A285348 Expansion of r(q^2) / r(q)^2 in powers of q where r() is the Rogers-Ramanujan continued fraction.

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A124242 Expansion of a parametrization of Ramanujan's continued fraction.

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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.

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