A200069 -id:A200069 - cp's OEIS frontend

A091533 Triangle read by rows, related to Pascal's triangle, starting with rows 1; 1,1.

1, 1, 1, 2, 3, 2, 3, 7, 7, 3, 5, 15, 21, 15, 5, 8, 30, 53, 53, 30, 8, 13, 58, 124, 157, 124, 58, 13, 21, 109, 273, 417, 417, 273, 109, 21, 34, 201, 577, 1029, 1239, 1029, 577, 201, 34, 55, 365, 1181, 2405, 3375, 3375, 2405, 1181, 365, 55, 89, 655, 2358, 5393, 8625, 10047, 8625, 5393, 2358, 655, 89

Offset: 0

Christian G. Bower, Jan 19 2004

T(n,k) is the number of lattice paths from (0,0) to (k,n-k) using steps (1,0),(2,0),(0,1),(0,2),(1,1). - Seiichi Manyama, Apr 26 2025.

			This triangle begins:
   1;
   1,   1;
   2,   3,    2;
   3,   7,    7,    3;
   5,  15,   21,   15,    5;
   8,  30,   53,   53,   30,     8;
  13,  58,  124,  157,  124,    58,   13;
  21, 109,  273,  417,  417,   273,  109,   21;
  34, 201,  577, 1029, 1239,  1029,  577,  201,   34;
  55, 365, 1181, 2405, 3375,  3375, 2405, 1181,  365,  55;
  89, 655, 2358, 5393, 8625, 10047, 8625, 5393, 2358, 655, 89;
  ...

Seiichi Manyama, Rows n = 0..139, flattened

Row sums: A015518(n+1). Columns 0-1: A000045(n+1), A023610(n-1).
Cf. A090174, A212338 (column 2), A192364 (central terms).
Cf. A036355.

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
     `if`(n<1, 1, add(add(T(n-i, k-j), j=0..i), i=1..2)))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jan 14 2022

A091533[-2, n2_] = 0; A091533[n1_, -2] = 0; A091533[-1, n2_] = 0; A091533[n1_, -1] = 0; A091533[0, 0] = 1; A091533[n1_, n2_] := A091533[n1, n2] = A091533[n1 - 1, n2] + A091533[n1, n2 - 1] + A091533[n1 - 1, n2 - 1] + A091533[n1 - 2, n2] + A091533[n1, n2 - 2]; Table[A091533[x - y, y], {x, 0, 9}, {y, 0, x}] // Flatten (* Robert P. P. McKone, Jan 14 2022 *)

T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) + T(n-2, k-1) + T(n-2, k-2) for n >= 2, k >= 0, with initial conditions specified by first two rows.
G.f.: A(x, y) = 1/(1-x-x*y-x^2-x^2*y-x^2*y^2).
Sum_{k = 0..n} T(n,k)*x^k = A000045(n+1), A015518(n+1), A015524(n+1), A200069(n+1) for x = 0, 1, 2, 3 respectively. - Philippe Deléham, Oct 30 2013
Sum_{k = 0..floor(n/2)} T(n-k,k) = (-1)^n*A079926(n). - Philippe Deléham, Oct 30 2013

A228815 Symmetric triangle, read by rows, related to Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 10, 14, 10, 3, 5, 20, 36, 36, 20, 5, 8, 38, 83, 106, 83, 38, 8, 13, 71, 182, 281, 281, 182, 71, 13, 21, 130, 382, 690, 834, 690, 382, 130, 21, 34, 235, 778, 1606, 2268, 2268, 1606, 778, 235, 34, 55, 420, 1546, 3586, 5780, 6750

Offset: 0

Philippe Deléham, Oct 30 2013

Triangles satisfying the same recurrence: A091533, A091562, A185081, A205575, A209137, A209138.

			Triangle begins :
0
1, 1
1, 2, 1
2, 5, 5, 2
3, 10, 14, 10, 3
5, 20, 36, 36, 20, 5
8, 38, 83, 106, 83, 38, 8
13, 71, 182, 281, 281, 182, 71, 13
21, 130, 382, 690, 834, 690, 382, 130, 21
34, 235, 778, 1606, 2268, 2268, 1606, 778, 235, 34
55, 420, 1546, 3586, 5780, 6750, 5780, 3586, 1546, 420, 55

Cf. A000045 (1st column), A001629 (2nd column), A008998, A152011, A261055 (3rd column).

G.f.: x*(1+y)/(1-x-x*y-x^2-x^2*y-x^2*y^2).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 0, T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
Sum_{k = 0..n} T(n,k)*x^k = A000045(n), 2*A015518(n), 3*A015524(n), 4*A200069(n) for x = 0, 1, 2, 3 respectively.
Sum_{k = 0..floor(n/2)} T(n-k,k) = A008998(n+1).

cp's OEIS Frontend

A091533 Triangle read by rows, related to Pascal's triangle, starting with rows 1; 1,1.

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