cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A285350 Convolution square of A285349.

Original entry on oeis.org

1, -4, 12, -24, 36, -36, 8, 56, -148, 228, -228, 80, 248, -688, 1048, -1048, 420, 896, -2588, 3920, -3884, 1648, 2864, -8488, 12744, -12516, 5440, 8400, -25176, 37480, -36504, 16032, 22892, -69024, 101968, -98504, 43468, 58736, -177584, 260416, -249732, 110328
Offset: 0

Views

Author

Seiichi Manyama, Apr 17 2017

Keywords

Comments

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

Crossrefs

Cf. A007325, A078905 (r(q)^5), A112274 (k(q)), A285349.

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

Views

Author

Seiichi Manyama, Apr 17 2017

Keywords

Comments

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).

Crossrefs

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).

Formula

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

A285628 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, 6, -6, 0, 12, -24, 27, -15, -12, 48, -81, 90, -54, -36, 159, -258, 267, -138, -123, 441, -684, 693, -354, -318, 1122, -1701, 1668, -801, -792, 2616, -3876, 3753, -1782, -1776, 5778, -8451, 8046, -3705, -3843, 12120, -17496, 16506, -7524, -7848, 24483
Offset: 0

Views

Author

Seiichi Manyama, Apr 22 2017

Keywords

Crossrefs

r(q)^k / r(q^k): A285349 (k=2), this sequence (k=3), A285629 (k=4), A285630 (k=5).
Cf. A285583.

A285629 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, 10, -16, 16, -4, -20, 48, -66, 60, -18, -64, 168, -248, 236, -80, -208, 536, -750, 688, -252, -528, 1432, -2048, 1908, -724, -1356, 3648, -5104, 4680, -1820, -3088, 8510, -12000, 11044, -4368, -6940, 19112, -26632, 24304, -9734, -14584, 40656, -56784, 51840
Offset: 0

Views

Author

Seiichi Manyama, Apr 22 2017

Keywords

Crossrefs

r(q)^k / r(q^k): A285349 (k=2), A285628 (k=3), this sequence (k=4), A285630 (k=5).
Cf. A285584.

A285630 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, 15, -30, 40, -25, -35, 140, -250, 285, -150, -210, 740, -1230, 1330, -675, -880, 3015, -4830, 5025, -2450, -3135, 10380, -16180, 16450, -7875, -9785, 31850, -48720, 48600, -22800, -27985, 89465, -134760, 132530, -61400, -74205, 234515, -349000, 339145
Offset: 0

Views

Author

Seiichi Manyama, Apr 22 2017

Keywords

Comments

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

Crossrefs

r(q)^k / r(q^k): A285349 (k=2), A285628 (k=3), A285629 (k=4), this sequence (k=5).
Cf. A078905 (u^5), A229793 (1 / u^5), A285585, A285587.

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

Views

Author

Seiichi Manyama, Apr 19 2017

Keywords

Comments

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.

Crossrefs

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

Programs

  • Ruby
    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)
Showing 1-6 of 6 results.