OFFSET
1,4
COMMENTS
Also the Bell transform of (-n)^n adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 16 2016
LINKS
D. H. Lehmer, Numbers Associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.
FORMULA
(k-1)!*a(n, k) = Sum_{i=0..k-1}((-1)^(n-k-i)*binomial(k-1, i)*(n-i-1)^(n-1)).
a(n,k) = (-1)^(n-k)*T(k,n-k,n-k), n>=k, where T(n,k,m)=m*T(n,m-1,k)+T(n-1,k,m+1), T(n,0,m)=1. - Vladimir Kruchinin, Mar 07 2020
EXAMPLE
The triangle T(n, k) begins:
[1] 1;
[2] -1, 1;
[3] 4, -3, 1;
[4] -27, 19, -6, 1;
[5] 256, -175, 55, -10, 1;
[6] -3125, 2101, -660, 125, -15, 1;
[7] 46656, -31031, 9751, -1890, 245, -21, 1;
[8] -823543, 543607, -170898, 33621, -4550, 434, -28, 1;
MAPLE
R := proc(n, k, m) option remember;
if k < 0 or n < 0 then 0 elif k = 0 then 1 else
m*R(n, k-1, m) + R(n-1, k, m+1) fi end:
A039621 := (n, k) -> (-1)^(n-k)*R(k-1, n-k, n-k):
MATHEMATICA
a[1, 1] = 1; a[n_, k_] := 1/(k-1)! Sum[((-1)^(n-k-i)*Binomial[k-1, i]*(n-i-1)^(n-1)), {i, 0, k-1}];
Table[a[n, k], {n, 1, 10}, {k, 1, n}]//Flatten (* Jean-François Alcover, Jun 03 2019 *)
PROG
(PARI) tabl(nn) = {for (n = 1, nn, for (k = 1, n, print1(sum(i = 0, k-1, (-1)^(n-k-i)*binomial(k-1, i)*(n-i-1)^(n-1))/(k-1)!, ", "); ); print(); ); } \\ Michel Marcus, Aug 28 2013
(Sage) # uses[bell_matrix from A264428]
# Adds 1, 0, 0, 0, ... as column 0 at the left side of the triangle.
bell_matrix(lambda n: (-n)^n, 7) # Peter Luschny, Jan 16 2016
(Maxima)
T(n, k, m):=if k<0 or n<0 then 0 else if k=0 then 1 else m*T(n, k-1, m)+T(n-1, k, m+1);
a(n, k):=if n<k then 0 else (-1)^(n-k)*T(k, n-k, n-k); /* Vladimir Kruchinin, Mar 07 2020
CROSSREFS
KEYWORD
tabl,sign
AUTHOR
STATUS
approved