A110197 Number triangle of sums of squared binomial coefficients.

1, 2, 1, 3, 5, 1, 4, 14, 10, 1, 5, 30, 46, 17, 1, 6, 55, 146, 117, 26, 1, 7, 91, 371, 517, 251, 37, 1, 8, 140, 812, 1742, 1476, 478, 50, 1, 9, 204, 1596, 4878, 6376, 3614, 834, 65, 1, 10, 285, 2892, 11934, 22252, 19490, 7890, 1361, 82, 1, 11, 385, 4917, 26334, 66352, 82994, 51990, 15761, 2107, 101, 1

Offset: 0

Author

Paul Barry, Jul 15 2005

Keywords

• easy
• nonn
• tabl

Comments

Alternatively, number square T(n,k) = Sum_{i=0..n} binomial(i+k,k)^2 read by antidiagonals.

Examples

			Rows start:
  1;
  2,   1;
  3,   5,   1;
  4,  14,  10,   1;
  5,  30,  46,  17,   1;
  6,  55, 146, 117,  26,   1;
  ...
		

Crossrefs

Columns include A000027, A000330, A024166, A086020, A086023, A086025.

Row sums are A006134.

Antidiagonal sums are A110198.

T(2n,n) gives A112029.

Programs

PARI

T(n,k) = sum(i=0, n-k, binomial(i+k,k)^2);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print();); \\ Michel Marcus, Dec 03 2016

Formula

T(n,k) = Sum_{i=0..n-k} binomial(i+k,k)^2.

G.f.: 1/((1-x)*sqrt(x^2*y^2-2*x^2*y-2*x*y+x^2-2*x+1)). - Vladimir Kruchinin, Mar 20 2025