A120406 Triangle read by rows: related to series expansion of the square root of 2 linear factors.

1, 2, 2, 5, 6, 5, 14, 18, 18, 14, 42, 56, 60, 56, 42, 132, 180, 200, 200, 180, 132, 429, 594, 675, 700, 675, 594, 429, 1430, 2002, 2310, 2450, 2450, 2310, 2002, 1430, 4862, 6864, 8008, 8624, 8820, 8624, 8008, 6864, 4862

Offset

0,2

Comments

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_{k=0..n} T(n,k)*a^k*b^(n-k))*(x/4)^n. (The g.f. below simply reformulates this fact.) A combinatorial interpretation of T(n,k) would be very interesting.

Links

Table of n, a(n) for n=0..44.

Formula

T(n,k) = 2*binomial(n,k)^2*binomial(2n+2,n)/binomial(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_{j=0..k-1} (k-j)*CatalanNumber(j)*CatalanNumber(n-j). This shows that T(n,k) is an integer. Generating function F(x,y):=Sum_{n>=0, k=0..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).

1 + x*(d/dx)(log(F(x,y))) = 1 + (2 + 2*y)*x + (6 + 4*y + 6*y^2)*x^2 + ... is the o.g.f. for A067804. - Peter Bala, Jul 17 2015

G.f. A(x,y) = -G(-x,y), G(x,y) satisfies G(x,y) = x/A008459(G(x,y))^2. - Vladimir Kruchinin, Oct 24 2020

Example

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

Mathematica

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

Prog

(Maxima)
solve(A=x*(A^2*y^2-2*A^2*y-2*A*y+A^2-2*A+1), A); /* Vladimir Kruchinin, oct 24 2020 */

Crossrefs

Column k=0 is the Catalan numbers A000108 (offset). The middle-of-row entries form A005566. Cf. A067804.

Sequence in context: A112573 A233740 A266595 * A050157 A209503 A209744

Adjacent sequences: A120403 A120404 A120405 * A120407 A120408 A120409

Keyword

nonn,tabl

Author

David Callan, Jul 03 2006

Status

approved