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%I #15 Dec 09 2019 10:36:35
%S 1,1,1,1,2,1,1,4,3,1,1,8,8,4,1,1,17,20,13,5,1,1,38,50,38,19,6,1,1,89,
%T 126,107,63,26,7,1,1,216,322,296,196,96,34,8,1,1,539,834,814,588,326,
%U 138,43,9,1,1,1374,2187,2236,1728,1052,507,190,53,10,1
%N Riordan array (1/(1-x),xm(x)) where m(x) is the g.f. of Motzkin numbers A001006.
%C Diagonal sums : A082395.
%H Alois P. Heinz, <a href="/A167630/b167630.txt">Rows n = 0..200, flattened</a>
%F T(n,0)=1, T(0,k)=0 for k>0, T(n,k)=0 if k>n, T(n,k)=T(n-1,k-1)+T(n-1,k)+T(n-1,k+1).
%F Sum_{k=0..n} k * T(n,k) = A003462(n). - _Alois P. Heinz_, Apr 20 2018
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 4, 3, 1;
%e 1, 8, 8, 4, 1;
%e 1, 17, 20, 13, 5, 1;
%e 1, 38, 50, 38, 19, 6, 1;
%e ...
%p T:= proc(n, k) option remember; `if`(k=0, 1,
%p `if`(k>n, 0, T(n-1, k-1)+T(n-1, k)+T(n-1, k+1)))
%p end:
%p seq(seq(T(n, k), k=0..n), n=0..12); # _Alois P. Heinz_, Apr 20 2018
%t T[_, 0] = T[n_, n_] = 1;
%t T[n_, k_] /; 0<k<n := T[n, k] = T[n-1, k-1] + T[n-1, k] + T[n-1, k+1];
%t T[_, _] = 0;
%t Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 09 2019 *)
%Y Diagonals include: A006416, A034856, A086615, A140662.
%Y Cf. A001006, A003462, A064189, A094531.
%K nonn,tabl
%O 0,5
%A _Philippe Deléham_, Nov 07 2009
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