A107105 Triangle, read by rows, where T(n,k) = C(n,k)*(C(n,k) + 1)/2, n>=k>=0.

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 21, 10, 1, 1, 15, 55, 55, 15, 1, 1, 21, 120, 210, 120, 21, 1, 1, 28, 231, 630, 630, 231, 28, 1, 1, 36, 406, 1596, 2485, 1596, 406, 36, 1, 1, 45, 666, 3570, 8001, 8001, 3570, 666, 45, 1, 1, 55, 1035, 7260, 22155, 31878, 22155, 7260

Offset: 0

Author

Paul D. Hanna, May 21 2005

Keywords

• nonn
• tabl
• easy

Comments

Replace each number x in Pascal's triangle by x(x+1)/2. - Charlie Marion, May 31 2013

Examples

			Triangle begins:
1;
1,1;
1,3,1;
1,6,6,1;
1,10,21,10,1;
1,15,55,55,15,1;
1,21,120,210,120,21,1;
1,28,231,630,630,231,28,1; ...
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A005317 (row sums), A107597 (antidiagonal sums).

Programs

Mathematica

Table[Binomial[n,k] (Binomial[n,k]+1)/2,{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Jul 20 2025 *)

PARI

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

Formula

n-th row sum equals A005317(n) = (C(2n, n) + 2^n)/2.