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Triangle of Lehmer-Comtet numbers of 2nd kind.
6

%I #41 Jun 11 2022 03:47:50

%S 1,-1,1,4,-3,1,-27,19,-6,1,256,-175,55,-10,1,-3125,2101,-660,125,-15,

%T 1,46656,-31031,9751,-1890,245,-21,1,-823543,543607,-170898,33621,

%U -4550,434,-28,1,16777216,-11012415,3463615,-688506,95781,-9702,714,-36,1

%N Triangle of Lehmer-Comtet numbers of 2nd kind.

%C 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

%H D. H. Lehmer, <a href="http://dx.doi.org/10.1216/RMJ-1985-15-2-461">Numbers Associated with Stirling Numbers and x^x</a>, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.

%F (k-1)!*a(n, k) = Sum_{i=0..k-1}((-1)^(n-k-i)*binomial(k-1, i)*(n-i-1)^(n-1)).

%F 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

%e The triangle T(n, k) begins:

%e [1] 1;

%e [2] -1, 1;

%e [3] 4, -3, 1;

%e [4] -27, 19, -6, 1;

%e [5] 256, -175, 55, -10, 1;

%e [6] -3125, 2101, -660, 125, -15, 1;

%e [7] 46656, -31031, 9751, -1890, 245, -21, 1;

%e [8] -823543, 543607, -170898, 33621, -4550, 434, -28, 1;

%p R := proc(n, k, m) option remember;

%p if k < 0 or n < 0 then 0 elif k = 0 then 1 else

%p m*R(n, k-1, m) + R(n-1, k, m+1) fi end:

%p A039621 := (n, k) -> (-1)^(n-k)*R(k-1, n-k, n-k):

%p seq(seq(A039621(n, k), k = 1..n), n = 1..9); # _Peter Luschny_, Jun 10 2022 after _Vladimir Kruchinin_

%t 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}];

%t Table[a[n, k], {n, 1, 10}, {k, 1, n}]//Flatten (* _Jean-François Alcover_, Jun 03 2019 *)

%o (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

%o (Sage) # uses[bell_matrix from A264428]

%o # Adds 1,0,0,0,... as column 0 at the left side of the triangle.

%o bell_matrix(lambda n: (-n)^n, 7) # _Peter Luschny_, Jan 16 2016

%o (Maxima)

%o 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);

%o a(n,k):=if n<k then 0 else (-1)^(n-k)*T(k,n-k,n-k); /* _Vladimir Kruchinin_, Mar 07 2020

%Y A008296 (matrix inverse), A354794 (variant), A045531 (column |a(n, 2)|).

%Y Cf. A185164.

%K tabl,sign

%O 1,4

%A _Len Smiley_