OFFSET
0,2
COMMENTS
There is a relationship to the left factorials A003422 (see formula).
FORMULA
T(n,n) = 1 for n >= 0; T(n,n-1) = n+2 for n > 0.
T(n,k) = (n+2) * (T(n-1,k) - T(n-2,k)) for 0 < k+1 < n.
T(n,k) + T(n-2,k) - 2 * T(n-1,k) = (n+1)! / ((k+2) * k!) for 0 < k+1 < n.
T(n,k) * T(n-1,k-1) - T(n,k-1) * T(n-1,k) = (n+2)! / (k+2)! for 0 < k < n.
The row polynomials p(n; x) = Sum_{k=0..n} T(n,k) * x^k satisfy the recurrence equation p(n; x) = (n+2) * (p(n-1; x) - p(n-2; x)) + x^n for n > 1 with p(0; x) = 1 and p(1; x) = 3 + x.
Row sums are p(n; 1): 1, 4, 13, 46, 199, 1072, ....
Alternating row sums are p(n; -1): 1, 2, 5, 14, 55, 286, ....
With T as lower triangular matrix the matrix inverse M is defined by: M(n,n) = 1 for n >= 0, M(n,n-1) = -(n+2) for n > 0, M(n,n-2) = n+2 for n > 1, and M(i,j) = 0 otherwise.
EXAMPLE
The triangle T(n,k), 0 <= k <= n, begins:
n\k : 0 1 2 3 4 5 6 7 8 9
====================================================================
0 : 1
1 : 3 1
2 : 8 4 1
3 : 25 15 5 1
4 : 102 66 24 6 1
5 : 539 357 133 35 7 1
6 : 3496 2328 872 232 48 8 1
7 : 26613 17739 6651 1773 369 63 9 1
8 : 231170 154110 57790 15410 3210 550 80 10 1
9 : 2250127 1500081 562529 150007 31251 5357 781 99 11 1
etc.
The matrix inverse M(n,k), 0 <= k <= n, begins:
n\k : 0 1 2 3 4 5 6 7 8 9
===============================================
0 : 1
1 : -3 1
2 : 4 -4 1
3 : 0 5 -5 1
4 : 0 0 6 -6 1
5 : 0 0 0 7 -7 1
6 : 0 0 0 0 8 -8 1
7 : 0 0 0 0 0 9 -9 1
8 : 0 0 0 0 0 0 10 -10 1
9 : 0 0 0 0 0 0 0 11 -11 1
etc.
MATHEMATICA
T[n_, k_] := (n + 2) * Sum[i!, {i, k, n}]/((k + 2)*k!); Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 12 2021 *)
CROSSREFS
KEYWORD
AUTHOR
Werner Schulte, May 16 2021
STATUS
approved