A192575 - OEIS

%I #17 Oct 13 2025 23:09:47

%S 1,2,0,2,0,-1,2,0,-3,0,2,0,-5,0,1,2,0,-7,0,4,0,2,0,-9,0,9,0,-1,2,0,

%T -11,0,16,0,-5,0,2,0,-13,0,25,0,-14,0,1,2,0,-15,0,36,0,-30,0,6,0,2,0,

%U -17,0,49,0,-55,0,20,0,-1

%N Triangle T(n,0) = A040000(n), T(n,k) = 0 (odd-numbered columns); T(n,k) = (-1)^(k/2)*A110813(n-k/2-1,k/2-1) (even-numbered columns, k>0).

%C A zero-padded variant of A110813, which provides more information.

%F T(n,k) = T(n-1,k)-T(n-2,k-2), n>1.

%F T(n,2*k+1) = 0.

%F T(n,2*k) = (-1)^k*binomial(n-k-1,k-1)*(2*n-3*k)/k, k>0. - _R. J. Mathar_, Aug 26 2011

%F T(n,0) = A040000(n).

%F Sum_{k=0..n} T(n,k) = A057079(n).

%F Sum_{k=0..n} |T(n,k)| = A000045(n+2). (See A129710).

%e Triangle begins:

%e   1;

%e   2 0;

%e   2 0  -1;

%e   2 0  -3 0;

%e   2 0  -5 0  1;

%e   2 0  -7 0  4 0;

%e   2 0  -9 0  9 0  -1;

%e   2 0 -11 0 16 0  -5 0;

%e   2 0 -13 0 25 0 -14 0 1;

%e   2 0 -15 0 36 0 -30 0 6 0;

%Y Cf. A000045, A040000, A057079, A110813, A129710, A191662.

%K sign,tabl,easy

%O 0,2

%A _Paul Curtz_, Jul 04 2011