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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.
0
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).
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
KEYWORD
nonn,tabl
AUTHOR
Ludovic Schwob, Nov 16 2020
EXTENSIONS
More terms from Stefano Spezia, Nov 17 2020
STATUS
approved