0,2

The numbers T(n,k) arise in the expansion of the square root of 2 generic linear factors: 1-Sqrt[(1-a*x)(1-b*x)] = (a+b)*x/2 + 1/8*(b-a)^2*x^2*Sum_{n>=0}( Sum_{0<=k<=n}T(n,k)*a^k*b^(n-k) )*(x/4)^n. (The GF below simply reformulates this fact.) A combinatorial interpretation of T(n,k) would be very interesting.

T(n,k) = 2 binom(n,k)^2 binom(2n+2,n)/binom(2n+2,2k+1). This shows that T(n,k) is positive and the rows are symmetric. T(n,k) = (k+1) CatalanNumber[n+1] - 2 Sum[(k-j)CatalanNumber[j]CatalanNumber[n-j],{j,0,k-1}]. This shows that T(n,k) is an integer. Generating function F(x,y):=Sum_{n>=0,0<=k<=n}T(n,k) x^n y^k is given by F(x,y) = ( 1-2x-2x*y-Sqrt[1-4x]Sqrt[1-4x*y] )/( 2x^2(1-y)^2 ). This shows that the row sums are the powers of 4 (A000302) because lim_{y->1}F(x,y) = 1/(1-4x).

Table begins

\ k..0....1....2....3....4....5....6

n

0 |..1

1 |..2....2

2 |..5....6....5

3 |.14...18...18...14

4 |.42...56...60...56...42

5 |132..180..200..200..180..132

6 |429..594..675..700..675..594..429

Table[2 Binomial[n, k]^2 Binomial[2n+2, n]/ Binomial[2n+2, 2k+1], {n, 0, 9}, {k, 0, n}]

Column k=0 is the Catalan numbers A000108 (offset). The middle-of-row entries form A005566.

Sequence in context: A000403 A068763 A112573 this_sequence A050157 A054255 A063177

Adjacent sequences: A120403 A120404 A120405 this_sequence A120407 A120408 A120409

nonn,tabf

David Callan (callan(AT)stat.wisc.edu), Jul 03 2006