%I A064189
%S A064189 1,1,1,2,2,1,4,5,3,1,9,12,9,4,1,21,30,25,14,5,1,51,76,69,44,20,6,1,127,
%T A064189 196,189,133,70,27,7,1,323,512,518,392,230,104,35,8,1,835,1353,1422,
%U A064189 1140,726,369,147,44,9,1,2188,3610,3915,3288,2235,1242,560,200,54,10,1
%N A064189 Triangle T(n,k), 0<=k<=n, read by rows, defined by: T(0,0)=1, T(n,k)= 
               0 if n<k, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1).
%C A064189 Motzkin triangle read in reverse order.
%C A064189 T(n,k) = number of lattice paths from (0,0) to (n,k), staying weakly 
               above the x-axis and consisting of steps U=(1,1), D=(1,-1) and H=(1,
               0). Example: T(3,1) = 5 because we have HHU, UDU, HUH, UHH and UUD. 
               Columns 0,1,2 and 3 give A001006 (Motzkin numbers), A002026 (first 
               differences of Motzkin numbers), A005322 and A005323, respectively. 
               - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 29 2004
%C A064189 Riordan array ((1-x-sqrt(1-2x-3x^2))/(2x^2), (1-x-sqrt(1-2x-3x^2))/(2x)). 
               Inverse is the array (1/(1+x+x^2), x/(1+x+x^2)) (A104562). - Paul 
               Barry (pbarry(AT)wit.ie), Mar 15 2005
%C A064189 Inverse binomial matrix applied to A039598 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Feb 28 2007
%C A064189 Triangle T(n,k), 0<=k<=n, read by rows given by : T(0,0)=1, T(n,k)=0 
               if k<0 or if k>n, T(n,0)=T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+T(n-1,
               k)+T(n-1,k+1) for k>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Mar 27 2007
%C A064189 This triangle belongs to the family of triangles defined by: T(0,0)=1, 
               T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,
               k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1 . Other triangles arise by choosing 
               different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,
               2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; 
               (1,2) -> A039599; (1,3) -> A110877; ((1,4) -> A124576; (2,0) -> A126075; 
               (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; 
               (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; 
               (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; 
               (5,5) -> A125906 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 
               25 2007
%C A064189 Equals binomial transform of triangle A053121 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Oct 25 2008]
%D A064189 See A026300 for references and other information.
%D A064189 E. Barcucci, R. Pinzani, R. Sprugnoli, The Motzkin family, P.U.M.A. Ser. 
               A, Vol. 2, 1991, No. 3-4, pp. 249-279.
%F A064189 Sum_{k=0..n} T(n, k)*(k+1) = 3^n.
%F A064189 Sum_{k=0..n} T(n, k)*T(n, n-k) = T(2*n, n) -T(2*n, n+2)
%F A064189 G.f. = M/(1-tzM), where M=1+zM+z^2M^2 is the g.f. of the Motzkin numbers 
               (A001006). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 29 2004
%F A064189 Sum_{k>=0} T(m, k)*T(n, k) = A001006(m+n) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Mar 05 2004
%F A064189 Sum_{k>=0} T(n-k, k) = A005043(n+2) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               May 31 2005
%F A064189 Column k has e.g.f. exp(x)*(BesselI(k,2x)-BesselI(k+2,2x)); - Paul Barry 
               (pbarry(AT)wit.ie), Feb 16 2006
%F A064189 T(n,k)=sum{j=0..n, C(n,j)*(C(n-j,j+k)-C(n-j,j+k+2))}; - Paul Barry (pbarry(AT)wit.ie), 
               Feb 16 2006
%F A064189 n-th row is generated from M^n * V, where M = the infinite tridiagonal 
               matrix with all 1's in the super, main and subdiagonals; and V = 
               the infinite vector [1,0,0,0...]. E.g. Row 3 = (4, 5, 3, 1), since 
               M^3 * V = [4, 5, 3, 1, 0, 0, 0...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Nov 04 2006
%F A064189 T(n,k)=A122896(n+1,k+1) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Apr 21 2007
%e A064189 1; 1,1; 2,2,1; 4,5,3,1; 9,12,9,4,1; ...
%Y A064189 Triangle in A026300 (the main entry for this sequence) with rows read 
               in reverse order.
%Y A064189 Cf. A001006, A002026, A005322, A005323.
%Y A064189 A053121 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 25 2008]
%Y A064189 Sequence in context: A001404 A104580 A105306 this_sequence A063415 A098977 
               A113547
%Y A064189 Adjacent sequences: A064186 A064187 A064188 this_sequence A064190 A064191 
               A064192
%K A064189 nonn,easy,tabl
%O A064189 0,4
%A A064189 N. J. A. Sloane (njas(AT)research.att.com), Sep 21 2001
%E A064189 More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 23 2001