cp's OEIS Frontend

A039621 Triangle of Lehmer-Comtet numbers of 2nd kind.

%I A039621 #41 Jun 11 2022 03:47:50
%S A039621 1,-1,1,4,-3,1,-27,19,-6,1,256,-175,55,-10,1,-3125,2101,-660,125,-15,
%T A039621 1,46656,-31031,9751,-1890,245,-21,1,-823543,543607,-170898,33621,
%U A039621 -4550,434,-28,1,16777216,-11012415,3463615,-688506,95781,-9702,714,-36,1
%N A039621 Triangle of Lehmer-Comtet numbers of 2nd kind.
%C A039621 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 A039621 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 A039621 (k-1)!*a(n, k) = Sum_{i=0..k-1}((-1)^(n-k-i)*binomial(k-1, i)*(n-i-1)^(n-1)).
%F A039621 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 A039621 The triangle T(n, k) begins:
%e A039621 [1]       1;
%e A039621 [2]      -1,      1;
%e A039621 [3]       4,     -3,       1;
%e A039621 [4]     -27,     19,      -6,     1;
%e A039621 [5]     256,   -175,      55,   -10,     1;
%e A039621 [6]   -3125,   2101,    -660,   125,   -15,   1;
%e A039621 [7]   46656, -31031,    9751, -1890,   245, -21,   1;
%e A039621 [8] -823543, 543607, -170898, 33621, -4550, 434, -28, 1;
%p A039621 R := proc(n, k, m) option remember;
%p A039621    if k < 0  or n < 0 then 0 elif k = 0 then 1 else
%p A039621    m*R(n, k-1, m) + R(n-1, k, m+1) fi end:
%p A039621 A039621 := (n, k) -> (-1)^(n-k)*R(k-1, n-k, n-k):
%p A039621 seq(seq(A039621(n, k), k = 1..n), n = 1..9); # _Peter Luschny_, Jun 10 2022 after _Vladimir Kruchinin_
%t A039621 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 A039621 Table[a[n, k], {n, 1, 10}, {k, 1, n}]//Flatten (* _Jean-François Alcover_, Jun 03 2019 *)
%o A039621 (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 A039621 (Sage) # uses[bell_matrix from A264428]
%o A039621 # Adds 1,0,0,0,... as column 0 at the left side of the triangle.
%o A039621 bell_matrix(lambda n: (-n)^n, 7) # _Peter Luschny_, Jan 16 2016
%o A039621 (Maxima)
%o A039621 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 A039621 a(n,k):=if n<k then 0 else  (-1)^(n-k)*T(k,n-k,n-k); /* _Vladimir Kruchinin_, Mar 07 2020
%Y A039621 A008296 (matrix inverse), A354794 (variant), A045531 (column |a(n, 2)|).
%Y A039621 Cf. A185164.
%K A039621 tabl,sign
%O A039621 1,4
%A A039621 _Len Smiley_