A134636 Triangle formed by Pascal's rule given borders = 2n+1.

1, 3, 3, 5, 6, 5, 7, 11, 11, 7, 9, 18, 22, 18, 9, 11, 27, 40, 40, 27, 11, 13, 38, 67, 80, 67, 38, 13, 15, 51, 105, 147, 147, 105, 51, 15, 17, 66, 156, 252, 294, 252, 156, 66, 17, 19, 83, 222, 408, 546, 546, 408, 222, 83, 19, 21, 102, 305, 630, 954, 1092, 954, 630, 305, 102, 21

Offset: 0

Author

Gary W. Adamson, Nov 04 2007

Keywords

• nonn
• tabl

Comments

Row sums = A048487: (1, 6, 16, 36, 76, 156, ...).

Examples

			First few rows of the triangle:
   1;
   3,  3;
   5,  6,  5;
   7, 11, 11,  7;
   9, 18, 22, 18,  9;
  11, 27, 40, 40, 27, 11;
  13, 38, 67, 80, 67, 38, 13;
  ...
		

Links

• Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A007318, A048487, A051601, A051597.

Programs

Haskell

a134636 n k = a134636_tabl !! n !! k
a134636_row n = a134636_tabl !! n
a134636_tabl = iterate (\row -> zipWith (+) ([2] ++ row) (row ++ [2])) [1]
-- Reinhard Zumkeller, Nov 23 2012

Maple

T:= proc(n,k) option remember;
      `if`(k<0 or k>n, 0,
      `if`(k=0 or k=n, 2*n+1,
         T(n-1, k-1) + T(n-1, k) ))
    end:
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, May 26 2013

Mathematica

NestList[Append[Prepend[Map[Apply[Plus, #] &, Partition[#, 2, 1]], #[[1]] + 2], #[[1]] + 2] &, {1}, 10] // Grid  (* Geoffrey Critzer, May 26 2013 *)
T[n_, k_] := Binomial[n, k-1] + Binomial[n, k] + 2 Binomial[n, k+1] + Binomial[n, n-k+1];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 07 2021 *)

Formula

Triangle, given borders = (1, 3, 5, 7, 9, ...); apply Pascal's rule T(n,k) = T(n-1,k) P T(n-1,k-1).

T(n,k) = A051601(n,k) + A051597(n,k); T(n,k) mod 2 = A047999(n,k). - Reinhard Zumkeller, Nov 23 2012

Closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 19 2013

Extensions

Offset changed by Reinhard Zumkeller, Nov 23 2012