A176306 Triangle T(n,k) = 1 - A176304(k) - A176304(n-k) + A176304(n), read by rows.

1, 1, 1, 1, 0, 1, 1, 13, 13, 1, 1, 25, 38, 25, 1, 1, -185, -160, -160, -185, 1, 1, -779, -964, -952, -964, -779, 1, 1, 7497, 6718, 6520, 6520, 6718, 7497, 1, 1, 45907, 53404, 52612, 52402, 52612, 53404, 45907, 1, 1, -524629, -478722, -471238, -472042, -472042, -471238, -478722, -524629, 1

Offset: 0

Author

Roger L. Bagula, Apr 14 2010

Keywords

• sign
• tabl

Comments

Row sums are {1, 2, 2, 28, 90, -688, -4436, 41472, 356250, -3893260, -41666708, ...}.

Examples

			Triangle begins as:
  1;
  1,     1;
  1,     0,     1;
  1,    13,    13,     1;
  1,    25,    38,    25,     1;
  1,  -185,  -160,  -160,  -185,     1;
  1,  -779,  -964,  -952,  -964,  -779,     1;
  1,  7497,  6718,  6520,  6520,  6718,  7497,     1;
  1, 45907, 53404, 52612, 52402, 52612, 53404, 45907, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Programs

Magma

function b(n)
  if n eq 0 then return 0;
  else return (-1)^n*n*b(n-1) -1;
  end if; return b; end function;
[1+b(n)-b(k)-b(n-k): k in [0..n], n in [1..10]]; // G. C. Greubel, Nov 26 2019

Maple

A176304 := proc(n)
    if n = 0 then
        0;
    else
        (-1)^n*n*procname(n-1)-1 ;
    end if;
end proc:
A176306 := proc(n,m)
    1-A176304(m)-A176304(n-m)+A176304(n) ;
end proc: # R. J. Mathar, May 04 2013

Mathematica

b[n_]:= b[n] = If[n==0, 0, (-1)^n*n*b[n-1] -1];
T[n_, k_]:= T[n, k] = 1 - (b[k] +b[n-k] -b[n]);
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten

PARI

b(n) = if(n==0, 0, (-1)^n*n*b(n-1) -1);
T(n,k) = 1 + b(n) - b(k) - b(n-k); \\ G. C. Greubel, Nov 26 2019

Sage

@CachedFunction
def b(n):
    if (n==0): return 0
    else: return (-1)^n*n*b(n-1) -1
[[1+b(n)-b(k)-b(n-k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 26 2019

Formula

T(n,k) = T(n,n-k).