OFFSET
0,5
FORMULA
T(n, k) = Sum_{i=k..n} A131689(i, k) * n! / (n-i)!.
T(n, k) = n! * k! * (Sum_{i=0..n-k} A048993(n-i, k) / i!).
T(n, k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * A320031(n, i).
Conjecture: E.g.f. of column k is exp(t) * t^k * k! / (Prod_{i=0..k} (1 - i*t)).
Conjecture: Sum_{k=0..n} (-1)^(n-k) * T(n, k) = A000166(n).
EXAMPLE
Lower triangular array starts:
n\k : 0 1 2 3 4 5 6 7
==========================================================================
0 : 1
1 : 1 1
2 : 1 4 4
3 : 1 15 48 36
4 : 1 64 504 1008 576
5 : 1 325 5680 22680 31680 14400
6 : 1 1956 72060 510480 1304640 1382400 518400
7 : 1 13699 1036224 12233340 50823360 94046400 79833600 25401600
etc.
MATHEMATICA
T[n_, k_] := Sum[(-1)^(k - j)*Binomial[k, j]*HypergeometricPFQ[{1, -n}, {}, -j], {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Peter Luschny, Apr 12 2024 *)
PROG
(PARI) T(n, k) = if(k==0, 1, if(k > n, 0, n*k*(T(n-1, k-1) + T(n-1, k))))
CROSSREFS
KEYWORD
AUTHOR
Werner Schulte, Apr 11 2024
STATUS
approved