A119245 Triangle, read by rows, defined by: T(n,k) = (4*k+1)*binomial(2*n+1, n-2*k)/(2*n+1) for n >= 2*k >= 0.

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, 16796, 48450, 23256, 3705, 170, 1, 58786, 177650, 95931, 19019, 1309, 21, 208012, 653752, 389367, 92092, 8602, 252, 1

Offset: 0

Author

Paul D. Hanna, May 10 2006

Keywords

• nonn
• tabf

Comments

Closely related to triangle A118919.

Row n contains 1+floor(n/2) terms.

From Peter Bala, 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)

Examples

			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; ...
		

Links

• Zoran Sunic, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003). - _Peter Bala_, Mar 20 2009

Crossrefs

Cf. A119244 (eigenvector), A088218, A000108, A000344, A001392; A118919 (variant), A158483; A002057, A002894.

Programs

Mathematica

f1 = (1-Sqrt[1-4*x])/(2*x);
DeleteCases[CoefficientList[Normal@Series[f1/(1 - x^2*y*f1^4),{x,0,10},{y,0,5}],{x,y}],0,Infinity]//TableForm  (* Bradley Klee, Feb 26 2018 *)
Table[(1+4*k)/(n+1+2*k)*Binomial[2*n,n+2*k],{n,0,10},{k,0,Floor[n/2]}]//TableForm (* Bradley Klee, Feb 26 2018 *)

PARI

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

Formula

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

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..floor(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).

...

T(n,k) = (4*k+1)/(n+2*k+1)*binomial(2*n,n+2*k). Compare with A158483. - Peter Bala, Mar 20 2009

T(n,k) = A039599(n, 2*k). - Johannes W. Meijer, Sep 04 2013

A002894(n) = Sum_{k=0..floor(n/2)} (binomial(2k,k)^2)*(4^(n-2*k))*T(n,k). - Bradley Klee, Feb 26 2018