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 1-4 of 4 results.

A339565 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (1,1), (1,2), (2,1).

Original entry on oeis.org

1, 3, 17, 101, 627, 3999, 25955, 170571, 1131433, 7559301, 50795985, 342935689, 2324278669, 15804931797, 107775401349, 736723618773, 5046774983235, 34636814325087, 238114193665451, 1639378334244867, 11301978856210543, 78010917772099207, 539055832175992119
Offset: 0

Views

Author

Kent Mei, Dec 08 2020

Keywords

Links

Crossrefs

Cf. A000984, A001850, A137635, A339390, A382436.

Programs

  • Maple
    a:= proc(n) local t; 1/(1-x-y-x*y-(x*y^2)-(x^2*y));
      for t in [x, y] do coeftayl(%, t=0, n) od
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 09 2020
# second Maple program:
b:= proc(l) option remember; `if`(l[2]=0, 1,
      add((f-> `if`(f[1]<0, 0, b(f)))(sort(l-h)), h=
      [[1, 0], [0, 1], [1$2], [1, 2], [2, 1]]))
    end:
a:= n-> b([n$2]):
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 09 2020
# third Maple program:
a:= proc(n) option remember; `if`(n<3, [1, 3, 17][n+1],
      ((6*n-3)*a(n-1)+(7*n-7)*a(n-2)+(4*n-6)*a(n-3))/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 09 2020
  • Mathematica
    b[l_] := b[l] = If[l[[2]] == 0, 1,
     Sum[Function[f, If[f[[1]] < 0, 0, b[f]]][Sort[l - h]], {h,
     {{1, 0}, {0, 1}, {1, 1}, {1, 2}, {2, 1}}}]];
a[n_] := b[{n, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 30 2022, after Alois P. Heinz *)

Formula

a(n) = [(x*y)^n] 1/(1-x-y-x*y-x*y^2-x^2*y). - Alois P. Heinz, Dec 09 2020
a(n) = A382436(2n,n). - Alois P. Heinz, Mar 25 2025
a(n) ~ sqrt((3776 + (26570110976 - 74946048*sqrt(177))^(1/3) + 8*(59*(879572 + 2481*sqrt(177)))^(1/3))/11328) * (2 + (459 - 12*sqrt(177))^(1/3)/3 + (153 + 4*sqrt(177))^(1/3)/3^(2/3))^n / sqrt(Pi*n). - Vaclav Kotesovec, Mar 26 2025

A382444 Triangle read by rows, defined by the two-variable g.f. (1 + y*x^2 + (y^2 + y)*x^3)/(1-(1+y)*x-y*x^2).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 9, 18, 9, 1, 1, 11, 34, 34, 11, 1, 1, 13, 54, 86, 54, 13, 1, 1, 15, 78, 174, 174, 78, 15, 1, 1, 17, 106, 306, 434, 306, 106, 17, 1, 1, 19, 138, 490, 914, 914, 490, 138, 19, 1, 1, 21, 174, 734, 1710, 2262, 1710, 734, 174, 21, 1
Offset: 0

Views

Author

F. Chapoton, Mar 25 2025

Keywords

Comments

Every row is symmetric.

Examples

			Triangle begins:
  [0] [1]
  [1] [1,  1]
  [2] [1,  4,   1]
  [3] [1,  7,   7,   1]
  [4] [1,  9,  18,   9,   1]
  [5] [1, 11,  34,  34,  11,   1]
  [6] [1, 13,  54,  86,  54,  13,   1]
  [7] [1, 15,  78, 174, 174,  78,  15,   1]
  [8] [1, 17, 106, 306, 434, 306, 106,  17,  1]
  [9] [1, 19, 138, 490, 914, 914, 490, 138, 19, 1]
  ...

Crossrefs

Similar to A008288, A103450 and A382436. Row sums are A265107.

Programs

  • Sage
    y = polygen(QQ, 'y')
x = y.parent()[['x']].gen()
gf = (1 + y*x^2 + (y^2 + y)*x^3)/(1 - (1 + y)*x - y*x^2)
[list(u) for u in list(gf.O(11))]

A382439 Triangle read by rows: defined by the two-variable g.f. (x^3*y^2 + x^3*y - x^2*y + 1) / (1 - x^2*y - x*y - x).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 7, 12, 7, 1, 1, 9, 24, 24, 9, 1, 1, 11, 40, 60, 40, 11, 1, 1, 13, 60, 124, 124, 60, 13, 1, 1, 15, 84, 224, 308, 224, 84, 15, 1, 1, 17, 112, 368, 656, 656, 368, 112, 17, 1, 1, 19, 144, 564, 1248, 1620, 1248, 564, 144, 19, 1
Offset: 0

Views

Author

F. Chapoton, Mar 25 2025

Keywords

Comments

The alternating sum of every row n > 0 vanishes. Every row is symmetric.

Examples

			  [0] [1]
  [1] [1,  1]
  [2] [1,  2,   1]
  [3] [1,  5,   5,   1]
  [4] [1,  7,  12,   7,   1]
  [5] [1,  9,  24,  24,   9,   1]
  [6] [1, 11,  40,  60,  40,  11,   1]
  [7] [1, 13,  60, 124, 124,  60,  13,   1]
  [8] [1, 15,  84, 224, 308, 224,  84,  15,  1]
  [9] [1, 17, 112, 368, 656, 656, 368, 112, 17, 1]

Crossrefs

Similar to A008288 and A382436. Row sums are A245990.

Programs

  • Sage
    y = polygen(QQ, 'y')
x = y.parent()[['x']].gen()
gf = (x^3*y^2 + x^3*y - x^2*y + 1)/(1 - x^2*y - x*y - x)
[list(u) for u in list(gf.O(10))]

A382448 Triangle read by rows, defined by the two-variable g.f. (x^3*y^2 + x^3*y + 1)/(1 - x^2*y - x*y - x).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 8, 15, 8, 1, 1, 10, 29, 29, 10, 1, 1, 12, 47, 73, 47, 12, 1, 1, 14, 69, 149, 149, 69, 14, 1, 1, 16, 95, 265, 371, 265, 95, 16, 1, 1, 18, 125, 429, 785, 785, 429, 125, 18, 1, 1, 20, 159, 649, 1479, 1941, 1479, 649, 159, 20, 1
Offset: 0

Views

Author

F. Chapoton, Mar 26 2025

Keywords

Comments

Every row is symmetric.

Examples

			Triangle begins:
  [0] [1]
  [1] [1,  1]
  [2] [1,  3,   1]
  [3] [1,  6,   6,   1]
  [4] [1,  8,  15,   8,   1]
  [5] [1, 10,  29,  29,  10,  1]
  [6] [1, 12,  47,  73,  47, 12,    1]
  [7] [1, 14,  69, 149, 149, 69,   14,   1]
  [8] [1, 16,  95, 265, 371, 265,  95,  16, 1]
  [9] [1, 18, 125, 429, 785, 785, 429, 125, 18, 1]

Crossrefs

Similar to A008288, A103450, A382436 and A382444. Row sums are A105082.

Programs

  • Sage
    y = polygen(QQ, 'y')
x = y.parent()[['x']].gen()
gf = (x^3*y^2 + x^3*y + 1)/(1 - x^2*y - x*y - x)
[list(u) for u in list(gf.O(10))]
Showing 1-4 of 4 results.

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