A285348 -id:A285348 - cp's OEIS frontend

A285349 Expansion of r(q)^2 / r(q^2) in powers of q where r() is the Rogers-Ramanujan continued fraction.

1, -2, 4, -4, 2, 2, -8, 12, -12, 6, 8, -24, 36, -36, 16, 20, -62, 92, -88, 40, 46, -144, 208, -196, 88, 102, -308, 440, -412, 180, 208, -624, 884, -816, 356, 404, -1206, 1692, -1552, 672, 760, -2244, 3128, -2852, 1224, 1378, -4048, 5612, -5084, 2174, 2428, -7104, 9796, -8836, 3760

Offset: 0

Seiichi Manyama, Apr 17 2017

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

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), A285628 (k=3), A285629 (k=4), A285630 (k=5).

Cf. A007325, A078905 (r(q)^5), A112274 (k(q)), A285348.

a(n) = A138518(n) + A285348(n) for n>0 (conjectured). - Thomas Baruchel, May 14 2018

A285585 Expansion of r(q^5) / r(q)^5 in powers of q where r() is the Rogers-Ramanujan continued fraction.

Original entry on oeis.org

1, 5, 10, 5, -15, -25, 10, 60, 25, -110, -150, 85, 360, 155, -505, -675, 330, 1410, 555, -1925, -2450, 1210, 4920, 1930, -6275, -7875, 3710, 15000, 5720, -18575, -22800, 10735, 42310, 15960, -50605, -61400, 28280, 110610, 41100, -129570, -155250, 71060, 274320

Offset: 0

Seiichi Manyama, Apr 22 2017

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G.f. A(q) satisfies: A(q) = v / u^5 = (v^4 + 2*v^3 + 4*v^2 + 3*v + 1) / (v^4 - 3*v^3 + 4*v^2 - 2*v + 1), where u = r(q) and v = r(q^5).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Bruce C. Berndt, Heng Huat Chan, Sen-Shan Huang, Soon-Yi Kang, Jaebum Sohn, and Seung Hwan Son, The Rogers-Ramanujan continued fraction, Journal of Computational and Applied Mathematics, Vol. 105, No. 1-2 (1999), 9-24.

r(q^k) / r(q)^k: A285348 (k=2), A285583 (k=3), A285584 (k=4), this sequence (k=5).

Cf. A078905 (u^5), A229793 (1 / u^5), A285587, A285630.

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.

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, 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.

A285583 Expansion of r(q^3) / r(q)^3 in powers of q where r() is the Rogers-Ramanujan continued fraction.

Original entry on oeis.org

1, 3, 3, -3, -9, -3, 15, 18, -12, -42, -12, 63, 72, -45, -153, -51, 195, 228, -123, -435, -144, 540, 621, -321, -1140, -393, 1332, 1536, -747, -2700, -924, 3084, 3528, -1683, -6063, -2097, 6714, 7668, -3549, -12843, -4425, 14004, 15894, -7263, -26208, -9057

Offset: 0

Seiichi Manyama, Apr 22 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..10000

r(q^k) / r(q)^k: A285348 (k=2), this sequence (k=3), A285584 (k=4), A285585 (k=5).

Cf. A055102, A285443, A285628.

A285584 Expansion of r(q^4) / r(q)^4 in powers of q where r() is the Rogers-Ramanujan continued fraction.

Original entry on oeis.org

1, 4, 6, 0, -12, -12, 12, 32, 2, -60, -54, 64, 152, 24, -228, -224, 180, 488, 94, -688, -680, 528, 1448, 336, -1884, -1932, 1276, 3744, 944, -4680, -4828, 3088, 9154, 2464, -10980, -11520, 6744, 20792, 5832, -24304, -25618, 14584, 45424, 13184, -51696, -54972

Offset: 0

Seiichi Manyama, Apr 22 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..10000

r(q^k) / r(q)^k: A285348 (k=2), A285583 (k=3), this sequence (k=4), A285585 (k=5).

Cf. A055103, A285444, A285629.

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

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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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A285349 Expansion of r(q)^2 / r(q^2) in powers of q where r() is the Rogers-Ramanujan continued fraction.

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A285585 Expansion of r(q^5) / r(q)^5 in powers of q where r() is the Rogers-Ramanujan continued fraction.

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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.

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A285583 Expansion of r(q^3) / r(q)^3 in powers of q where r() is the Rogers-Ramanujan continued fraction.

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A285584 Expansion of r(q^4) / r(q)^4 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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