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 1-2 of 2 results.

A382436 Triangle read by rows, defined by the two-variable g.f. 1/(1 - (y + 1)*x - y*x^2 - (y^2 + y)*x^3).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 9, 17, 9, 1, 1, 12, 36, 36, 12, 1, 1, 15, 64, 101, 64, 15, 1, 1, 18, 101, 227, 227, 101, 18, 1, 1, 21, 147, 440, 627, 440, 147, 21, 1, 1, 24, 202, 767, 1459, 1459, 767, 202, 24, 1, 1, 27, 266, 1235, 2994, 3999, 2994, 1235, 266, 27, 1
Offset: 0

Views

Author

F. Chapoton, Mar 25 2025

Keywords

Comments

The original definition was "Decomposition of A077938".
Every row is symmetric.

Examples

			Triangle begins:
  1;
  1,  1;
  1,  3,   1;
  1,  6,   6,    1;
  1,  9,  17,    9,    1;
  1, 12,  36,   36,   12,    1;
  1, 15,  64,  101,   64,   15,    1;
  1, 18, 101,  227,  227,  101,   18,    1;
  1, 21, 147,  440,  627,  440,  147,   21,   1;
  1, 24, 202,  767, 1459, 1459,  767,  202,  24,  1;
  1, 27, 266, 1235, 2994, 3999, 2994, 1235, 266, 27, 1;
  ...

Crossrefs

Similar to A008288, A103450, and A382444.
Row sums are A077938.
T(2n, n) gives A339565.
Cf. A056594.

Programs

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

Formula

G.f. 1/(1 - (y + 1)*x - y*x^2 - (y^2 + y)*x^3).
Sum_{k=0..n} (-1)^k * T(n,k) = A056594(n). - Alois P. Heinz, Mar 25 2025

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 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.