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A225372 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 = -2.

%I A225372 #17 Mar 17 2022 23:46:59
%S A225372 1,1,1,1,-2,1,1,-1,-1,1,1,-4,6,-4,1,1,-3,2,2,-3,1,1,-6,15,-20,15,-6,1,
%T A225372 1,-5,9,-5,-5,9,-5,1,1,-8,28,-56,70,-56,28,-8,1,1,-7,20,-28,14,14,-28,
%U A225372 20,-7,1,1,-10,45,-120,210,-252,210,-120,45,-10,1
%N A225372 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 = -2.
%H A225372 G. C. Greubel, <a href="/A225372/b225372.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A225372 T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), with T(n, 1) = T(n, n) = 1, and m = -2.
%F A225372 Sum_{k=1..n} T(n, k) = A130706(n-1). - _G. C. Greubel_, Mar 17 2022
%e A225372 Triangle begins:
%e A225372   1;
%e A225372   1,  1;
%e A225372   1, -2,  1;
%e A225372   1, -1, -1,   1;
%e A225372   1, -4,  6,  -4,  1;
%e A225372   1, -3,  2,   2, -3,   1;
%e A225372   1, -6, 15, -20, 15,  -6,   1;
%e A225372   1, -5,  9,  -5, -5,   9,  -5,  1;
%e A225372   1, -8, 28, -56, 70, -56,  28, -8,  1;
%e A225372   1, -7, 20, -28, 14,  14, -28, 20, -7, 1;
%p A225372 T:=proc(n,k,l) option remember;
%p A225372 if (n=1 or k=1 or k=n) then 1 else
%p A225372 (l*n-l*k+1)*T(n-1,k-1,l)+(l*k-l+1)*T(n-1,k,l); fi; end;
%p A225372 for n from 1 to 14 do lprint([seq(T(n,k,-2),k=1..n)]); od;
%t A225372 T[n_, k_, l_] := T[n, k, l] = If[n == 1 || k == 1 || k == n, 1, (l*n-l*k+1)*T[n-1, k-1, l]+(l*k-l+1)*T[n-1, k, l]]; Table[T[n, k, -2], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jan 09 2014, translated from Maple *)
%o A225372 (Magma)
%o A225372 function T(n,k,m)
%o A225372   if k eq 1 or k eq n then return 1;
%o A225372   else return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m);
%o A225372   end if; return T;
%o A225372 end function;
%o A225372 A225372:= func< n,k | T(n,k,-2) >;
%o A225372 [A225372(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 17 2022
%o A225372 (Sage)
%o A225372 @CachedFunction
%o A225372 def T(n,k,m):
%o A225372     if (k==1 or k==n): return 1
%o A225372     else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
%o A225372 def A225372(n,k): return T(n,k,-2)
%o A225372 flatten([[ A225372(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Mar 17 2022
%Y A225372 For m = ...,-2,-1,0,1,2,3,4,5,6,7,8, ... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, A142459, A142560, A142561, A142562, A167884, ...
%Y A225372 Cf. A130706 (row sums).
%K A225372 sign,tabl
%O A225372 1,5
%A A225372 _N. J. A. Sloane_ and _Roger L. Bagula_, May 08 2013