0,5

Row sums are A014137 (partial sums of Catalan numbers A000108). Columns equal the partial sums of the columns of the Catalan convolution triangle A033184. Columns include A014137, A014138, A001453.

Apart from the first column, any term is the partial sum of terms of the row above, when summing from the right. - R. Stephan, Apr 27 2004

Matrix inverse equals triangle A104402.

T(n, k) = sum_{j=0..n-k} T(n-k, j)*C(k+j-1, k-1).

Recurrence: for k>0, T(n, k) = sum[j=k..n, T(n-1, j)]. - R. Stephan, Apr 27 2004

G.f.: 2/(2-y*(1-sqrt(1-4*x)))/(1-x). T(n, k) = T(n-1, k-1) + T(n, k+1) for n>0, with T(0, 0)=1.

T(7,3) = 41 = (1)*1 + (4)*3 + (3)*6 + (1)*10

= T(4,0)*C(2,2)+T(4,1)*C(3,2)+T(4,2)*C(5,2)+T(4,3)*C(6,2).

Rows begin:

{1},

{1,1},

{1,2,1},

{1,4,3,1},

{1,9,8,4,1},

{1,23,22,13,5,1},

{1,65,64,41,19,6,1},

{1,197,196,131,67,26,7,1},

{1,626,625,428,232,101,34,8,1},

{1,2056,2055,1429,804,376,144,43,9,1},...

(PARI) {T(n, k)=if(k>n|n<0|k<0, 0, if(k==0|k==n, 1, sum(j=0, n-k, T(n-k, j)*binomial(k+j-1, k-1)); ); )}

(PARI) {T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); polcoeff(polcoeff(2/(2-Y*(1-sqrt(1-4*X)))/(1-X), n, x), k, y)} (PARI) {T(n, k)=if(n<k|k<0, 0, if(n==k|k==0, 1, T(n-1, k-1)+T(n, k+1)))}

Cf. A033184, A000108, A014137, A014138, A001453.

Cf. A104402.

Sequence in context: A112682 A033185 A105632 this_sequence A117418 A101494 A125781

Adjacent sequences: A091488 A091489 A091490 this_sequence A091492 A091493 A091494

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jan 14 2004