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

0,5

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).

Links

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

Example

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, ...
  ...

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 *)

Crossrefs

Central diagonal terms give A002893.

Antidiagonal sums give A097893.

Sequence in context: A108359 A100936 A086620 * A228356 A253670 A137897

Adjacent sequences: A338931 A338932 A338933 * A338935 A338936 A338937

Keyword

nonn,tabl

Author

Ludovic Schwob, Nov 16 2020

Extensions

More terms from Stefano Spezia, Nov 17 2020

Status

approved