A338934 Square array T(i,j) = Sum_{k=0...min(i,j)} C(i,k)*C(j,k)*C(2*k,k) (i>=0,j>=0), read by antidiagonals.

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 15, 7, 1, 1, 9, 31, 31, 9, 1, 1, 11, 53, 93, 53, 11, 1, 1, 13, 81, 213, 213, 81, 13, 1, 1, 15, 115, 411, 639, 411, 115, 15, 1, 1, 17, 155, 707, 1551, 1551, 707, 155, 17, 1, 1, 19, 201, 1121, 3239, 4653, 3239, 1121, 201, 19, 1

Offset: 0

Author

Ludovic Schwob, Nov 16 2020

Keywords

• nonn
• tabl

Comments

T(i,j)*C(i+j,i) is the number of ways to write the vector (i,i,j,j) as a sum of vectors containing two occurrences of the number 1.

Up to order, the number of different sums is A106255(i+1,j+1).

Examples

			There are T(1,1)*C(2,1)=6 ways to write the vector (1,1,1,1) as a sum of vectors containing two occurrences of the number 1 : (1,1,0,0)+(0,0,1,1), (0,0,1,1)+(1,1,0,0), (1,0,1,0)+(0,1,0,1), (0,1,0,1)+(1,0,1,0), (1,0,0,1)+(0,1,1,0), (0,1,1,0)+(1,0,0,1).
The square array T(i,j) (i >= 0, j >= 0) begins:
  1,  1,  1,   1,    1,    1, ...
  1,  3,  5,   7,    9,   11, ...
  1,  5, 15,  31,   53,   81, ...
  1,  7, 31,  93,  213,  411, ...
  1,  9, 53, 213,  639, 1551, ...
  1, 11, 81, 411, 1551, 4653, ...
  ...
		

Crossrefs

Central diagonal terms give A002893.

Antidiagonal sums give A097893.

Programs

Mathematica

T[i_,j_]:=Sum[Binomial[i,k]Binomial[j,k]Binomial[2k,k],{k,0,Min[i,j]}]; Flatten[Table[T[i-j,j],{i,0,10},{j,0,i}]] (* Stefano Spezia, Nov 17 2020 *)

Extensions

More terms from Stefano Spezia, Nov 17 2020