A105868 Triangle read by rows, T(n,k) = C(n,k)*C(k,n-k).

1, 0, 1, 0, 2, 1, 0, 0, 6, 1, 0, 0, 6, 12, 1, 0, 0, 0, 30, 20, 1, 0, 0, 0, 20, 90, 30, 1, 0, 0, 0, 0, 140, 210, 42, 1, 0, 0, 0, 0, 70, 560, 420, 56, 1, 0, 0, 0, 0, 0, 630, 1680, 756, 72, 1, 0, 0, 0, 0, 0, 252, 3150, 4200, 1260, 90, 1, 0, 0, 0, 0, 0, 0, 2772, 11550, 9240, 1980, 110, 1, 0, 0

Offset: 0

Author

Paul Barry, Apr 23 2005

Keywords

• easy
• nonn
• tabl

Comments

Row sums are the central trinomial coefficients A002426.

Product of A007318 and this sequence is A008459.

Coefficient array for polynomials P(n,x) = x^n*F(1/2-n/2,-n/2;1;4/x). - Paul Barry, Oct 04 2008

Column sums give A001850. It appears that the sums along the antidiagonals of the triangle produce A182883. - Peter Bala, Mar 06 2013

Examples

			Triangle begins
  1;
  0,  1;
  0,  2,  1;
  0,  0,  6,  1;
  0,  0,  6, 12,  1;
  0,  0,  0, 30, 20, 1;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows

Crossrefs

Cf. A063007. A001850 (column sums), A182883.

Programs

Magma

[[Binomial(n,k)*Binomial(k,n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jun 14 2015

Maple

gf := 1/((1 - x*y)^2 - 4*y^2*x)^(1/2):
yser := series(gf, y, 12): ycoeff := n -> coeff(yser, y, n):
row := n -> seq(coeff(expand(ycoeff(n)), x, k), k=0..n):
seq(row(n), n=0..7); # Peter Luschny, Oct 28 2020

Mathematica

Flatten[Table[Binomial[n,k]Binomial[k,n-k],{n,0,20},{k,0,n}]] (* Harvey P. Dale, Nov 12 2014 *)

Formula

G.f.: 1/(sqrt((1-x*y)^2-4*x^2*y)). - Vladimir Kruchinin, Oct 28 2020