0,4

Motzkin triangle read in reverse order.

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

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

Inverse binomial matrix applied to A039598 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 28 2007

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

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

See A026300 for references and other information.

E. Barcucci, R. Pinzani, R. Sprugnoli, The Motzkin family, P.U.M.A. Ser. A, Vol. 2, 1991, No. 3-4, pp. 249-279.

Sum_{k=0..n} T(n, k)*(k+1) = 3^n.

Sum_{k=0..n} T(n, k)*T(n, n-k) = T(2*n, n) -T(2*n, n+2)

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

Sum_{k>=0} T(m, k)*T(n, k) = A001006(m+n) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 05 2004

Sum_{k>=0} T(n-k, k) = A005043(n+2) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 31 2005

Column k has e.g.f. exp(x)*(BesselI(k,2x)-BesselI(k+2,2x)); - Paul Barry (pbarry(AT)wit.ie), Feb 16 2006

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

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

T(n,k)=A122896(n+1,k+1) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 21 2007

1; 1,1; 2,2,1; 4,5,3,1; 9,12,9,4,1; ...

Triangle in A026300 (the main entry for this sequence) with rows read in reverse order.

Cf. A001006, A002026, A005322, A005323.

Adjacent sequences: A064186 A064187 A064188 this_sequence A064190 A064191 A064192

Sequence in context: A104580 A106197 A105306 this_sequence A063415 A098977 A113547

nonn,easy,tabl

njas, Sep 21 2001

More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 23 2001