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.

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, 1, -5, 9, -5, -5, 9, -5, 1, 1, -8, 28, -56, 70, -56, 28, -8, 1, 1, -7, 20, -28, 14, 14, -28, 20, -7, 1, 1, -10, 45, -120, 210, -252, 210, -120, 45, -10, 1

Offset: 1

Author

N. J. A. Sloane and Roger L. Bagula, May 08 2013

Keywords

• sign
• tabl

Examples

			Triangle begins:
  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;
  1, -5,  9,  -5, -5,   9,  -5,  1;
  1, -8, 28, -56, 70, -56,  28, -8,  1;
  1, -7, 20, -28, 14,  14, -28, 20, -7, 1;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Crossrefs

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, ...

Cf. A130706 (row sums).

Programs

Magma

function T(n,k,m)
  if k eq 1 or k eq n then return 1;
  else return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m);
  end if; return T;
end function;
A225372:= func< n,k | T(n,k,-2) >;
[A225372(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 17 2022

Maple

T:=proc(n,k,l) option remember;
if (n=1 or k=1 or k=n) then 1 else
(l*n-l*k+1)*T(n-1,k-1,l)+(l*k-l+1)*T(n-1,k,l); fi; end;
for n from 1 to 14 do lprint([seq(T(n,k,-2),k=1..n)]); od;

Mathematica

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 *)

Sage

@CachedFunction
def T(n,k,m):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
def A225372(n,k): return T(n,k,-2)
flatten([[ A225372(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 17 2022

Formula

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.

Sum_{k=1..n} T(n, k) = A130706(n-1). - G. C. Greubel, Mar 17 2022