0,3

Closely related to triangle A118919. Row n contains 1+floor(n/2) terms. Row sums equal A088218(n) = C(2*n-1,n). T(n,0)=A000108(n) (the Catalan numbers). T(n,1)=A000344(n). T(n,2)=A001392(n). Sum_{k=0..[n/2]} k*T(n,k) = A000346(n-2). Eigenvector is defined by: A119244(n) = Sum_{k=0..[n\2]} T(n,k)*A119244(k).

Contribution from Peter Bala (pbala(AT)talktalk.net), Mar 20 2009: (Start)

Combinatorial interpretations of T(n,k):

1) The number of standard tableaux of shape (n-2*k,n+2*k).

2) The entries in column k are (with an offset of 2*k) the number of n-th generation vertices in the tree of sequences with unit increase labeled by 4*k. See [Sunik, Theorem 4].

(End)

Zoran Sunik, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003). [From Peter Bala (pbala(AT)talktalk.net), Mar 20 2009]

G.f.: A(x,y) = C/(1-x*y^2*C^4), where C=[1-sqrt(1-4*x)]/(2*x) is the Catalan g.f. (A000108).

T(n,k) = (4*k+1)/(n+2*k+1)*binomial(2*n,n+2*k). Compare with A158483. [From Peter Bala (pbala(AT)talktalk.net), Mar 20 2009]

Triangle begins:

1;

1;

2, 1;

5, 5;

14, 20, 1;

42, 75, 9;

132, 275, 54, 1;

429, 1001, 273, 13;

1430, 3640, 1260, 104, 1;

4862, 13260, 5508, 663, 17; ...

(PARI) T(n, k)=(4*k+1)*binomial(2*n+1, n-2*k)/(2*n+1)

Cf. A119244 (eigenvector), A088218, A000108, A000344, A001392; A118919 (variant).

Cf. A158483. [From Peter Bala (pbala(AT)talktalk.net), Mar 20 2009]

Sequence in context: A074392 A052547 A096976 this_sequence A128731 A129157 A086905

Adjacent sequences: A119242 A119243 A119244 this_sequence A119246 A119247 A119248

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), May 10 2006