A104557 Triangle T, read by rows, such that the unsigned columns of the matrix inverse when read downwards equals the rows of T read backwards, with T(n,n)=1 and T(n,n-1) = floor((n+1)/2)*floor((n+2)/2).
1, 1, 1, 2, 2, 1, 6, 6, 4, 1, 24, 24, 18, 6, 1, 120, 120, 96, 36, 9, 1, 720, 720, 600, 240, 72, 12, 1, 5040, 5040, 4320, 1800, 600, 120, 16, 1, 40320, 40320, 35280, 15120, 5400, 1200, 200, 20, 1, 362880, 362880, 322560, 141120, 52920, 12600, 2400, 300, 25, 1
Offset: 0
Examples
Rows of T begin:
1;
1, 1;
2, 2, 1;
6, 6, 4, 1;
24, 24, 18, 6, 1;
120, 120, 96, 36, 9, 1;
720, 720, 600, 240, 72, 12, 1;
5040, 5040, 4320, 1800, 600, 120, 16, 1;
40320, 40320, 35280, 15120, 5400, 1200, 200, 20, 1; ...
The matrix inverse A104558 begins:
1;
-1, 1;
0, -2, 1;
0, 2, -4, 1;
0, 0, 6, -6, 1;
0, 0, -6, 18, -9, 1;
0, 0, 0, -24, 36, -12, 1;
0, 0, 0, 24, -96, 72, -16, 1; ...
Programs
-
PARI
T(n,k)=(n-k)!*binomial(n-(k\2),(k+1)\2)*binomial(n-((k+1)\2),k\2)
Formula
Formula: T(n,k) = (n-k)!*C(n-floor(k/2), floor((k+1)/2))*C(n-floor((k+1)/2), floor(k/2)).
Recurrence: T(n,k) = n*T(n-1,k) + T(n-2,k-2) for n >= k >= 2, with T(0,0) = T(1,0) = T(1,1) = 1.
T(n,0) = n!.
T(n,k) = T(n-1,k-1) + floor((k+2)/2)*T(n,k+1), T(0,0)=1, T(n,k)=0 for k > n or for k < 0. - Philippe Deléham, Dec 18 2006
Comments