%I A020474
%S A020474 1,0,1,0,1,2,0,0,2,4,0,0,1,5,9,0,0,0,3,12,21,0,0,0,1,9,30,51,0,0,0,0,4,
%T A020474 25,76,127,0,0,0,0,1,14,69,196,323,0,0,0,0,0,5,44,189,512,835,0,0,0,0,
%U A020474 0,1,20,133,518,1353,2188,0,0,0,0,0,0,6,70,392,1422,3610,5798,0,0,0,0
%N A020474 A Motzkin triangle: a(n,k), n >= 2, 1<=k<=n, = number of complete, strictly 
               subdiagonal staircase functions.
%C A020474 T(n,k) = number of Dyck n-paths that start UU, contain no DUDU and no 
               subpath of the form UUPDD with P a nonempty Dyck path and whose terminal 
               descent has length n-k+2. For example, T(5,4)=2 counts UUDUUDUDDD, 
               UUUDDUUDDD (each ending with exactly n-k+2=3 Ds). - David Callan 
               (callan(AT)stat.wisc.edu), Sep 25 2006
%D A020474 Martin Aigner, Motzkin numbers. European J. Combin. 19 (1998), 663-675.
%D A020474 J. L. Chandon, J. LeMaire and J. Pouget, Denombrement des quasi-ordres 
               sur un ensemble fini, Math. Sci. Humaines, No 62 (1978), 61-80.
%D A020474 R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series 
               A, 23 (1977), 291-301.
%D A020474 Paul Peart and Wen-jin Woan, A divisibility property for a subgroup of 
               Riordan matrices. Discrete Appl. Math. 98 (2000), 255-263.
%F A020474 a(n, k)=a(n, k-1)+a(n-1, k-1)+a(n-2, k-1), n>k >= 2.
%e A020474 1
%e A020474 0,1
%e A020474 0,1,2
%e A020474 0,0,2,4
%e A020474 0,0,1,5,9
%e A020474 0,0,0,3,12,21
%e A020474 0,0,0,1,9,30,51
%e A020474 0,0,0,0,4,25,76,127
%e A020474 0,0,0,0,1,14,69,196,323
%t A020474 a[2,2]=1; a[n_,k_]/;Not[n>2 && 2<=k<=n] := 0; a[n_,k_]/;(n>2 && 2<=k<=n) 
               := a[n,k] = a[n,k-1] + a[n-1,k-1] + a[n-2,k-1]; Table[a[n,k],{n,2,
               10},{k,2,n}] - David Callan (callan(AT)stat.wisc.edu), Sep 25 2006
%o A020474 (PARI) T(n,k)=if(n==0&&k==0,1,if(n<=0||k<=0||n<k,0,T(n,k-1)+T(n-1,k-1)+T(n-2,
               k-1))) (from R. Stephan)
%Y A020474 Main diagonal is A001006.
%Y A020474 Other diagonals include A002026, A005322, A005323, A005324, A005325. 
               Row sums are (essentially) A005043.
%Y A020474 Sequence in context: A151669 A115509 A134312 this_sequence A135589 A158122 
               A028641
%Y A020474 Adjacent sequences: A020471 A020472 A020473 this_sequence A020475 A020476 
               A020477
%K A020474 nonn,tabl,easy,nice
%O A020474 2,6
%A A020474 N. J. A. Sloane (njas(AT)research.att.com).
%E A020474 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 04 2000