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A167630 Riordan array (1/(1-x),xm(x)) where m(x) is the g.f. of Motzkin numbers A001006.

%I A167630 #23 May 02 2025 16:23:13
%S A167630 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 A167630 126,107,63,26,7,1,1,216,322,296,196,96,34,8,1,1,539,834,814,588,326,
%U A167630 138,43,9,1,1,1374,2187,2236,1728,1052,507,190,53,10,1
%N A167630 Riordan array (1/(1-x),xm(x)) where m(x) is the g.f. of Motzkin numbers A001006.
%H A167630 Alois P. Heinz, <a href="/A167630/b167630.txt">Rows n = 0..200, flattened</a>
%F A167630 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 A167630 Sum_{k=0..n} k * T(n,k) = A003462(n). - _Alois P. Heinz_, Apr 20 2018
%F A167630 Sum_{k=0..n} (-1)^(k+1) * T(n,k) = A082397(n-2) for n>=2. - _Alois P. Heinz_, May 02 2025
%e A167630 Triangle begins:
%e A167630   1;
%e A167630   1,  1;
%e A167630   1,  2,  1;
%e A167630   1,  4,  3,  1;
%e A167630   1,  8,  8,  4,  1;
%e A167630   1, 17, 20, 13,  5, 1;
%e A167630   1, 38, 50, 38, 19, 6, 1;
%e A167630   ...
%p A167630 T:= proc(n, k) option remember; `if`(k=0, 1,
%p A167630       `if`(k>n, 0, T(n-1, k-1)+T(n-1, k)+T(n-1, k+1)))
%p A167630     end:
%p A167630 seq(seq(T(n, k), k=0..n), n=0..12);  # _Alois P. Heinz_, Apr 20 2018
%t A167630 T[_, 0] = T[n_, n_] = 1;
%t A167630 T[n_, k_] /; 0<k<n := T[n, k] = T[n-1, k-1] + T[n-1, k] + T[n-1, k+1];
%t A167630 T[_, _] = 0;
%t A167630 Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 09 2019 *)
%Y A167630 Antidiagonal sums give A082395.
%Y A167630 Row sums give A383527.
%Y A167630 Diagonals include: A006416, A034856, A086615, A140662.
%Y A167630 Cf. A001006, A003462, A064189, A082397, A094531.
%K A167630 nonn,tabl
%O A167630 0,5
%A A167630 _Philippe Deléham_, Nov 07 2009