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A081114 Triangle read by rows: T(n,k) = n*T(n-1,k) + n - k starting at T(n,n)=0.

%I A081114 #12 Jun 17 2019 01:39:01
%S A081114 0,1,0,4,1,0,15,5,1,0,64,23,6,1,0,325,119,33,7,1,0,1956,719,202,45,8,
%T A081114 1,0,13699,5039,1419,319,59,9,1,0,109600,40319,11358,2557,476,75,10,1,
%U A081114 0,986409,362879,102229,23019,4289,679,93,11,1,0,9864100,3628799,1022298,230197,42896,6795,934,113,12,1,0
%N A081114 Triangle read by rows: T(n,k) = n*T(n-1,k) + n - k starting at T(n,n)=0.
%C A081114 Taking the triangle into negative values of n and k would produce results close to (k+1)*e*n! - 1, i.e., one less than multiples of A000522 for nonnegative n.
%F A081114 For k > 0, T(n, k) = ceiling((A001339(k-1)/(k-1)! - (k-1)*e) *n! - 1) where A001339(k-1) = ceiling((k-1)!*(k-1)*e) for k > 1.
%F A081114 T(n, 0) = floor(e*n! - 1) for n > 0; T(n, 1) = n! - 1. T(n, n)=0; T(n, n-1) = n+2; T(n, n-2) = n^2 + 3*n + 5 = A027688(n+1).
%e A081114 Triangle begins
%e A081114     0;
%e A081114     1,   0;
%e A081114     4,   1,  0;
%e A081114    15,   5,  1, 0;
%e A081114    64,  23,  6, 1, 0;
%e A081114   325, 119, 33, 7, 1, 0;
%o A081114 (PARI) T(n,k) = if (k==n, 0, n*T(n-1,k) + n - k);
%o A081114 tabl(nn) = {for (n=0, nn, for (k=0, n, print1(T(n, k), ", ");); print(););} \\ _Michel Marcus_, Jun 16 2019
%Y A081114 Columns include A007526 and A033312.
%K A081114 nonn,tabl
%O A081114 0,4
%A A081114 _Henry Bottomley_, Apr 16 2003
%E A081114 More terms from _Michel Marcus_, Jun 16 2019