OFFSET
0,5
COMMENTS
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n, n) = 1, T(n+1, n) = -(n+1) for n >= 0; otherwise T(n, k) = T(n-k, 0) = -A003319(n-k-1) for n > k+1 and k >= 0.
Sum_{k=0..n} T(n, k) = A104985(n). - G. C. Greubel, Jun 07 2021
EXAMPLE
Triangle begins:
1;
-1, 1;
-1, -2, 1;
-3, -1, -3, 1;
-13, -3, -1, -4, 1;
-71, -13, -3, -1, -5, 1;
-461, -71, -13, -3, -1, -6, 1;
-3447, -461, -71, -13, -3, -1, -7, 1;
-29093, -3447, -461, -71, -13, -3, -1, -8, 1; ...
MATHEMATICA
T[n_, k_]:= If[k==n, 1, If[k==n-1, -n, -A003319[n-k]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 07 2021 *)
PROG
(PARI) T(n, k)=if(n==k, 1, if(n==k+1, -n, -(n-k)!-sum(i=1, n-k-1, i!*T(n-k-i, 0))));
(Sage)
@CachedFunction
def T(n, k):
if (k==n): return 1
elif (k==n-1): return -n
else: return -factorial(n-k) - sum( factorial(j)*T(n-k-j, 0) for j in (1..n-k-1) )
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 07 2021
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Apr 10 2005
STATUS
approved