A039621 Triangle of Lehmer-Comtet numbers of 2nd kind.
1, -1, 1, 4, -3, 1, -27, 19, -6, 1, 256, -175, 55, -10, 1, -3125, 2101, -660, 125, -15, 1, 46656, -31031, 9751, -1890, 245, -21, 1, -823543, 543607, -170898, 33621, -4550, 434, -28, 1, 16777216, -11012415, 3463615, -688506, 95781, -9702, 714, -36, 1
Offset: 1
Examples
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;
Links
- D. H. Lehmer, Numbers Associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.
Programs
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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): seq(seq(A039621(n, k), k = 1..n), n = 1..9); # Peter Luschny, Jun 10 2022 after Vladimir Kruchinin
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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 *) -
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
Vladimir Kruchinin, Mar 07 2020 -
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
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
Comments