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Triangle array of binomial coefficients.
1

%I #16 Oct 29 2022 04:52:18

%S 1,1,1,1,1,2,1,1,2,3,1,1,3,3,6,1,1,3,4,6,10,1,1,4,4,10,10,20,1,1,4,5,

%T 10,15,20,35,1,1,5,5,15,15,35,35,70,1,1,5,6,15,21,35,56,70,126,1,1,6,

%U 6,21,21,56,56,126,126,252,1,1,6,7,21,28,56,84,126,210,252,462

%N Triangle array of binomial coefficients.

%F T(n, k) = binomial(floor((n+k)/2), floor(k/2)).

%e Triangle begins:

%e k=0 1 2 3 4 5

%e n=0: 1;

%e n=1: 1, 1;

%e n=2: 1, 1, 2;

%e n=3: 1, 1, 2, 3;

%e n=4: 1, 1, 3, 3, 6;

%e n=5: 1, 1, 3, 4, 6, 10;

%e ...

%t T[n_, k_]=Binomial[Floor[(n+k)/2], Floor[k/2]]; Table[T[n,k],{n,0,11},{k,0,n}] (* _Stefano Spezia_, Aug 23 2022 *)

%o (PARI) T(n, k) = binomial((n+k)\2, k\2); \\ _Michel Marcus_, Oct 29 2022

%Y Cf. A026010 (row sums), A016116 (diagonal sums), A001405 (main diagonal).

%K easy,nonn,tabl

%O 0,6

%A _Paul Barry_, Jul 15 2004