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

0,5

Comments

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

Links

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

Formula

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

Example

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; ...

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A162745 A001263 A162747 * A088925 A100862 A098568

Adjacent sequences: A107102 A107103 A107104 * A107106 A107107 A107108

Keyword

nonn,tabl,easy

Author

Paul D. Hanna, May 21 2005

Status

approved