A356692 Pascal-like triangle, where each entry is the sum of the four entries above it starting with 1 at the top.

1, 1, 1, 2, 2, 2, 4, 6, 6, 4, 10, 16, 20, 16, 10, 26, 46, 62, 62, 46, 26, 72, 134, 196, 216, 196, 134, 72, 206, 402, 618, 742, 742, 618, 402, 206, 608, 1226, 1968, 2504, 2720, 2504, 1968, 1226, 608, 1834, 3802, 6306, 8418, 9696, 9696, 8418, 6306, 3802, 1834, 5636, 11942, 20360, 28222, 34116, 36228, 34116, 28222, 20360, 11942, 5636

Offset

0,4

Comments

Similar in spirit to the regular Pascal triangle, except that here we have T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) + T(n-1,k+1), with the understanding that T(0,0) is defined to be 1, and T(n,k) is defined as 0 for k<0 and k>n.

T(n,k) is the number of permutations p of [n+1] such that at most one element of {p(1),...,p(i-1)} is between p(i) and p(i+1) for all i <= n and p(n+1) = k+1. T(4,1) = 16: 13542, 14532, 15342, 15432, 31542, 35142, 35412, 41532, 45132, 45312, 51342, 51432, 53142, 53412, 54132, 54312. - Alois P. Heinz, Aug 31 2022

Links

Alois P. Heinz, Rows n = 0..150, flattened

Formula

T(n,k) = T(n,n-k).

Example

T(4,0) = 10 because it is the sum of T(3,-2), T(3,-1), T(3,0), and T(3,1) which gives 0+0+4+6 = 10.
Triangle begins:
             1
           1   1
         2   2   2
       4   6   6   4
     10  16  20  16  10
   26  46  62  62  46  26
  ...

Maple

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, add(T(n-1, j), j=k-2..k+1)))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Aug 28 2022

Mathematica

T[0, 0] = 1; T[n_, k_] := T[n, k] = If[k < 0 || k > n, 0,
    T[n - 1, k - 2] + T[n - 1, k - 1] + T[n - 1, k] + T[n - 1, k + 1]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten

Crossrefs

Row sums give A216837(n+1).

Column k=0 and also main diagonal give A356832.

T(2n,n) gives A356853.

Cf. A007318, A064189.

Sequence in context: A328106 A342336 A320908 * A361404 A248781 A236840

Adjacent sequences: A356689 A356690 A356691 * A356693 A356694 A356695

Keyword

nonn,tabl

Author

Greg Dresden and Sadek Mohammed, Aug 23 2022

Status

approved