A338817 Matrix inverse of triangle A176270, read by rows.

1, -1, 1, -1, 0, 1, -3, 1, 1, 1, -12, 4, 5, 2, 1, -60, 20, 25, 11, 3, 1, -360, 120, 150, 66, 19, 4, 1, -2520, 840, 1050, 462, 133, 29, 5, 1, -20160, 6720, 8400, 3696, 1064, 232, 41, 6, 1, -181440, 60480, 75600, 33264, 9576, 2088, 369, 55, 7, 1, -1814400, 604800, 756000, 332640, 95760, 20880, 3690, 550, 71, 8, 1

Offset: 0

Author

Werner Schulte, Nov 10 2020

Keywords

• sign
• easy
• tabl

Examples

			The triangle T(n,k) for 0 <= k <= n starts:
n\k :       0      1      2      3     4     5    6   7  8  9
=============================================================
  0 :       1
  1 :      -1      1
  2 :      -1      0      1
  3 :      -3      1      1      1
  4 :     -12      4      5      2     1
  5 :     -60     20     25     11     3     1
  6 :    -360    120    150     66    19     4    1
  7 :   -2520    840   1050    462   133    29    5   1
  8 :  -20160   6720   8400   3696  1064   232   41   6  1
  9 : -181440  60480  75600  33264  9576  2088  369  55  7  1
etc.
		

Crossrefs

Cf. A000007, A176270.

Programs

PARI

for(n=0,10,for(k=0,n,if(k==n,print(" 1"),if(k==n-1,print1(n-2,", "),print1((k^2+k-1)*n!/(k+2)!,", ")))))

PARI

1/matrix(10, 10, n, k, n--; k--; if (n>=k, 1 + k*(k-n))) \\ Michel Marcus, Nov 11 2020

Formula

T(n,n) = 1 for n >= 0; T(n,n-1) = n - 2 for n > 0; T(n,n-2) = n^2 - 3*n + 1 for n > 1; T(n,k) = (k^2 + k - 1) * n! / (k+2)! for 0 <= k <= n-2.

T(n,k) = n * T(n-1,k) for 0 <= k < n-2.

T(n,k) = T(k+2,k) * n! / (k+2)! for 0 <= k <= n-2.

Row sums are A000007(n) for n >= 0.