A143418 Triangle read by rows. T(n,k) = binomial(n,k)*(binomial(n,k)-1)/2.

1, 3, 3, 6, 15, 6, 10, 45, 45, 10, 15, 105, 190, 105, 15, 21, 210, 595, 595, 210, 21, 28, 378, 1540, 2415, 1540, 378, 28, 36, 630, 3486, 7875, 7875, 3486, 630, 36, 45, 990, 7140, 21945, 31626, 21945, 7140, 990, 45, 55, 1485, 13530, 54285, 106491, 106491

Offset: 1

Author

Gary W. Adamson and Roger L. Bagula, Aug 14 2008

Keywords

• nonn
• easy
• tabl

Comments

Row sums = A108958: (1, 6, 27, 110, 430, 1652, ...).

Examples

			Row 4 of Pascal's triangle (1, 4, 6, 4, 1) with each term squared = (1, 16, 36, 16, 1), then subtracting (1, 4, 6, 4, 1) = (0, 12, 30, 12, 0). Dividing by 2 and deleting the zeros, we get row 4 of A143418: (6, 15, 6).
First few rows of the triangle =
1;
3, 3;
6, 15, 6;
10, 45, 45, 10;
15, 105, 190, 105, 15;
21, 210, 595, 595, 210, 21;
28, 378, 1540, 2415, 1540, 378, 28;
...
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A007318, A108958.

Programs

Maple

A143418 := proc(n,k)
        binomial(n,k)*(binomial(n,k)-1)/2 ;
end proc:
seq(seq(A143418(n,k),k=1..n-1),n=1..12) ; # R. J. Mathar, Apr 04 2012

Mathematica

Table[Binomial[n,k] (Binomial[n,k]-1)/2,{n,20},{k,n-1}]//Flatten (* Harvey P. Dale, Jun 14 2021 *)

Formula

T(n,k) = A065420(n-1,k-1)/2. - R. J. Mathar, Apr 04 2012

Extensions

Corrected by Harvey P. Dale, Jun 14 2021