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A144431 Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n,1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1)+(m*k-m+1)*T(n-1,k), where m = -1. 9

%I

%S 1,1,1,1,0,1,1,-1,-1,1,1,-2,2,-2,1,1,-3,2,2,-3,1,1,-4,7,-8,7,-4,1,1,

%T -5,9,-5,-5,9,-5,1,1,-6,16,-26,30,-26,16,-6,1,1,-7,20,-28,14,14,-28,

%U 20,-7,1

%N Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n,1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1)+(m*k-m+1)*T(n-1,k), where m = -1.

%C Row sums are: {1, 2, 2, 0, 0, 0, 0, 0, 0, 0, ...}.

%C For m = ...,-1,0,1,2 we get ..., A144431, A007318 (Pascal), A008292, A060187, ..., so this might be called a sub-Pascal triangle.

%C The triangle starts off like A098593, but is different further on.

%F m=-1; A(n,k) := (m*n - m*k + 1)*A(n-1, k-1) + (m*k - (m-1))*A(n-1, k).

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 0, 1;

%e 1, -1, -1, 1;

%e 1, -2, 2, -2, 1;

%e 1, -3, 2, 2, -3, 1;

%e 1, -4, 7, -8, 7, -4, 1;

%e 1, -5, 9, -5, -5, 9, -5, 1;

%e 1, -6, 16, -26, 30, -26, 16, -6, 1;

%e 1, -7, 20, -28, 14, 14, -28, 20, -7, 1;

%e ...

%p T:=proc(n,k,l) option remember;

%p if (n=1 or k=1 or k=n) then 1 else

%p (l*n-l*k+1)*T(n-1,k-1,l)+(l*k-l+1)*T(n-1,k,l); fi; end;

%p for n from 1 to 8 do lprint([seq(T(n,k,-1),k=1..n)]); od; # _N. J. A. Sloane_, May 08 2013

%t m=-1; A[n_, 1] := 1; A[n_, n_] := 1; A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k]; a = Table[A[n, k], {n, 10}, {k, n}]; Flatten[a]

%Y Cf. A007318, A008292, A060187, A098593.

%K tabl,sign

%O 1,12

%A _Roger L. Bagula_, Oct 04 2008

%E Edited by _N. J. A. Sloane_, May 08 2013

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Last modified May 26 02:56 EDT 2020. Contains 334613 sequences. (Running on oeis4.)