A373717 - OEIS

A373717

Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..floor(k*n/(2*k+1))} binomial(k * (n-2*j),j).

%I #15 Jun 15 2024 09:23:10

%S 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,3,3,1,1,1,1,4,5,4,1,1,1,1,5,7,8,

%T 6,1,1,1,1,6,9,13,15,9,1,1,1,1,7,11,19,28,26,13,1,1,1,1,8,13,26,45,53,

%U 45,19,1,1,1,1,9,15,34,66,91,105,80,28,1,1,1,1,10,17,43,91,141,201,211,140,41,1

%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..floor(k*n/(2*k+1))} binomial(k * (n-2*j),j).

%F G.f. of column k: 1/(1 - x * (1 + x^2)^k).

%F T(n,k) = Sum_{j=0..k} binomial(k,j) * T(n-2*j-1,k).

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 3, 4, 5, 6, 7, ...

%e 1, 3, 5, 7, 9, 11, 13, ...

%e 1, 4, 8, 13, 19, 26, 34, ...

%e 1, 6, 15, 28, 45, 66, 91, ...

%o (PARI) T(n, k) = sum(j=0, k*n\(2*k+1), binomial(k*(n-2*j), j));

%Y Columns k=0..3 give A000012, A000930, A193147, A373718.

%Y Main diagonal gives A373719.

%Y Cf. A099233.

%K nonn,tabl

%O 0,14

%A _Seiichi Manyama_, Jun 15 2024