Sadek Mohammed - cp's OEIS frontend

User: Sadek Mohammed

Sadek Mohammed has authored 1 sequences.

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

Greg Dresden and Sadek Mohammed, Aug 23 2022

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

			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
  ...

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

Row sums give A216837(n+1).
Column k=0 and also main diagonal give A356832.
T(2n,n) gives A356853.
Cf. A007318, A064189.

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

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

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

cp's OEIS Frontend

User: Sadek Mohammed

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

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