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A122851 Number triangle T(n,k) = C(k,n-k)*(n-k)!.

%I A122851 #47 Jun 12 2025 19:46:50
%S A122851 1,0,1,0,1,1,0,0,2,1,0,0,2,3,1,0,0,0,6,4,1,0,0,0,6,12,5,1,0,0,0,0,24,
%T A122851 20,6,1,0,0,0,0,24,60,30,7,1,0,0,0,0,0,120,120,42,8,1,0,0,0,0,0,120,
%U A122851 360,210,56,9,1,0,0,0,0,0,0,720,840,336,72,10,1
%N A122851 Number triangle T(n,k) = C(k,n-k)*(n-k)!.
%C A122851 Row sums are A122852.
%C A122851 Triangle T(n,k), read by rows, given by (0,1,-1,0,0,1,-1,0,0,1,-1,0,0,1,...) DELTA (1,0,0,-1,2,0,0,-2,3,0,0,-3,4,...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Dec 12 2011
%H A122851 Alois P. Heinz, <a href="/A122851/b122851.txt">Rows n = 0..140, flattened</a>
%F A122851 Number triangle T(n,k) = [k<=n]*k!/(2k-n)!.
%F A122851 T(n,k) = A008279(k,n-k). - _Danny Rorabaugh_, Apr 23 2015
%e A122851 Triangle begins
%e A122851   1;
%e A122851   0, 1;
%e A122851   0, 1, 1;
%e A122851   0, 0, 2, 1;
%e A122851   0, 0, 2, 3,  1;
%e A122851   0, 0, 0, 6,  4,  1;
%e A122851   0, 0, 0, 6, 12,  5, 1;
%e A122851   0, 0, 0, 0, 24, 20, 6, 1;
%e A122851   ...
%t A122851 Flatten[Table[Binomial[k,n-k](n-k)!,{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, May 16 2012 *)
%o A122851 (Magma) /* As triangle: */ [[Binomial(k,n-k)*Factorial(n-k): k in [0..n]]: n in [0.. 7]]; // _Vincenzo Librandi_, Apr 24 2015
%Y A122851 Cf. A094587, A242653.
%Y A122851 T(2n,n) gives A000142.
%K A122851 nonn,tabl,easy
%O A122851 0,9
%A A122851 _Paul Barry_, Sep 14 2006