A307090 - OEIS

A307090

Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,2*j) * binomial(n-k,2*j).

%I #20 May 20 2021 04:44:44

%S 1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,-2,-2,1,1,1,1,-5,-8,-5,1,1,1,1,-9,

%T -17,-17,-9,1,1,1,1,-14,-29,-34,-29,-14,1,1,1,1,-20,-44,-54,-54,-44,

%U -20,1,1,1,1,-27,-62,-74,-74,-74,-62,-27,1,1,1,1,-35,-83,-90,-74,-74,-90,-83,-35,1,1

%N Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,2*j) * binomial(n-k,2*j).

%H Seiichi Manyama, <a href="/A307090/b307090.txt">Rows n = 0..139, flattened</a>

%e Triangle begins:

%e n\k | 0 1 2 3 4 5 6 7 8

%e ----+-------------------------------------

%e 0 | 1;

%e 1 | 1, 1;

%e 2 | 1, 1, 1;

%e 3 | 1, 1, 1, 1;

%e 4 | 1, 1, 0, 1, 1;

%e 5 | 1, 1, -2, -2, 1, 1;

%e 6 | 1, 1, -5, -8, -5, 1, 1;

%e 7 | 1, 1, -9, -17, -17, -9, 1, 1;

%e 8 | 1, 1, -14, -29, -34, -29, -14, 1, 1;

%t T[n_, k_] := Sum[(-1)^j * Binomial[k, 2*j] * Binomial[n - k, 2*j], {j, 0, n - k}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 20 2021 *)

%Y Row sums give A099587(n+1).

%Y T(2*n,n) gives A307091.

%Y Cf. A007318, A098593, A119326, A119335.

%K sign,tabl,look

%O 0,18

%A _Seiichi Manyama_, Mar 24 2019