OFFSET
0,4
FORMULA
T(n,k) = T(n,k-1) + k * T(n-1,k-1) for 0 < k <= n with initial values T(n,0) = n for n >= 0 and T(i,j) = 0 if j < 0 or j > i.
T(n,k) = k! + T(n-1,k-1) * (k+1) for 0 < k <= n.
T(n,k) = (k+1)! + T(n-1,k) for 0 <= k < n.
E.g.f. of main diagonal (case n=0) and n-th subdiagonal (n>0): Sum_{k>=0} T(n+k,k) * x^k / k! = (n - log(1-x)) / (1-x)^2 for n >= 0.
G.f. of column k>=0: Sum_{n>=k} T(n,k) * y^n = (T(k,k) * y^k + ((k+1)! - T(k,k)) * y^(k+1)) / (1-y)^2.
G.f.: Sum_{n>=0, k=0..n} T(n,k)*x^k*y^n/k! = (y - (1-y) * log(1-x*y)) / ((1-y)^2 * (1-x*y)^2).
EXAMPLE
The triangle starts:
n\k : 0 1 2 3 4 5 6 7 8 9
=================================================================
0 : 0
1 : 1 1
2 : 2 3 5
3 : 3 5 11 26
4 : 4 7 17 50 154
5 : 5 9 23 74 274 1044
6 : 6 11 29 98 394 1764 8028
7 : 7 13 35 122 514 2484 13068 69264
8 : 8 15 41 146 634 3204 18108 109584 663696
9 : 9 17 47 170 754 3924 23148 149904 1026576 6999840
...
CROSSREFS
KEYWORD
AUTHOR
Werner Schulte, Aug 02 2020
STATUS
approved