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A034928 Triangle of ballot numbers.

%I A034928 #24 Jul 08 2025 21:28:22
%S A034928 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,
%T A034928 36,51,51,1,1,7,12,30,54,91,127,127,1,1,8,14,39,75,142,232,323,323,1,
%U A034928 1,9,16,49,99,205,370,603,835,835,1,1,10,18,60,126,281,545
%N A034928 Triangle of ballot numbers.
%H A034928 Reinhard Zumkeller, <a href="/A034928/b034928.txt">Rows n = 0..125 of triangle, flattened</a>
%H A034928 M. Aigner, <a href="http://dx.doi.org/10.1006/eujc.1998.0235">Motzkin Numbers</a>, Europ. J. Comb. 19 (1998), 663-675.
%F A034928 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).
%F A034928 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.
%e A034928 Triangle begins
%e A034928 1, 1,
%e A034928 1, 1, 1,
%e A034928 1, 1, 2, 2,
%e A034928 1, 1, 3, 4, 4,
%e A034928 1, 1, 4, 6, 9, 9,
%e A034928 1, 1, 5, 8, 15, 21, 21,
%e A034928 1, 1, 6, 10, 22, 36, 51, 51,
%e A034928 1, 1, 7, 12, 30, 54, 91, 127, 127,
%e A034928 1, 1, 8, 14, 39, 75, 142, 232, 323, 323,
%e A034928 1, 1, 9, 16, 49, 99, 205, 370, 603, 835, 835,
%e A034928 ...
%t A034928 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_) *)
%o A034928 (Haskell)
%o A034928 a034928 n k = a034928_tabf !! n !! k
%o A034928 a034928_row n = a034928_tabf !! n
%o A034928 a034928_tabf = iterate f [1,1] where
%o A034928    f us = vs ++ [last vs] where
%o A034928           vs = zipWith (+) us (0 : scanl (+) 0 us)
%o A034928 -- _Reinhard Zumkeller_, Sep 20 2014
%Y A034928 Right-hand edge is A001006.
%Y A034928 Cf. A247364 (mirrored).
%K A034928 nonn,tabf,easy
%O A034928 0,8
%A A034928 _N. J. A. Sloane_
%E A034928 More terms from _David W. Wilson_.