A034928 Triangle of ballot numbers.
1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 4, 4, 1, 1, 4, 6, 9, 9, 1, 1, 5, 8, 15, 21, 21, 1, 1, 6, 10, 22, 36, 51, 51, 1, 1, 7, 12, 30, 54, 91, 127, 127, 1, 1, 8, 14, 39, 75, 142, 232, 323, 323, 1, 1, 9, 16, 49, 99, 205, 370, 603, 835, 835, 1, 1, 10, 18, 60, 126, 281, 545
Offset: 0
Examples
Triangle begins 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 4, 4, 1, 1, 4, 6, 9, 9, 1, 1, 5, 8, 15, 21, 21, 1, 1, 6, 10, 22, 36, 51, 51, 1, 1, 7, 12, 30, 54, 91, 127, 127, 1, 1, 8, 14, 39, 75, 142, 232, 323, 323, 1, 1, 9, 16, 49, 99, 205, 370, 603, 835, 835, ...
Links
- Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
- M. Aigner, Motzkin Numbers, Europ. J. Comb. 19 (1998), 663-675.
Programs
-
Haskell
a034928 n k = a034928_tabf !! n !! k a034928_row n = a034928_tabf !! n a034928_tabf = iterate f [1,1] where f us = vs ++ [last vs] where vs = zipWith (+) us (0 : scanl (+) 0 us) -- Reinhard Zumkeller, Sep 20 2014 -
Mathematica
a[n_, 0] := 1; a[n_, 1] := 1; a[n_, 2] := n; a[n_, k_] := If [k > n + 1, 0, a[n - 1, k] + a[n, k - 1] + a[n - 1, k - 2] - a[n - 1, k - 1]]; Grid[Table[a[n, k], {n, 0, 10}, {k, 0, n + 1}]] (* Replace Grid with Flatten to get the sequence. *) (* L. Edson Jeffery, Aug 02 2014 (after David W. Wilson) *)
Formula
a(0, 0)=a(0, 1)=1, a(n, n+1)=a(n, n), a(n, k)=a(n-1, 0)+...+a(n-1, k-2)+a(n-1, k) (n >= 1, 0<=k<=n).
Or, from David W. Wilson: a(n, 0) = 1; a(n, 1) = 1; a(n, 2) = n; a(n, k) = 0 if k > n+1; a(n, k) = a(n-1, k) + a(n, k-1) + a(n-1, k-2) - a(n-1, k-1) otherwise.
Extensions
More terms from David W. Wilson.