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

0,2

Comments

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

Links

Table of n, a(n) for n=0..65.

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

Example

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

Prog

(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

Crossrefs

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

Row sums are A006134.

Antidiagonal sums are A110198.

T(2n,n) gives A112029.

Sequence in context: A104029 A208752 A119308 * A124819 A124019 A337886

Adjacent sequences: A110194 A110195 A110196 * A110198 A110199 A110200

Keyword

easy,nonn,tabl

Author

Paul Barry, Jul 15 2005

Status

approved