A176307 Triangle T(n,k) = 1 + A176304(k) + A176304(n-k) - A176304(n), read by rows.

1, 1, 1, 1, 2, 1, 1, -11, -11, 1, 1, -23, -36, -23, 1, 1, 187, 162, 162, 187, 1, 1, 781, 966, 954, 966, 781, 1, 1, -7495, -6716, -6518, -6518, -6716, -7495, 1, 1, -45905, -53402, -52610, -52400, -52610, -53402, -45905, 1, 1, 524631, 478724, 471240, 472044, 472044, 471240, 478724, 524631, 1

Offset: 0

Author

Roger L. Bagula, Apr 14 2010

Keywords

• sign
• tabl

Comments

Row sums are {1, 2, 4, -20, -80, 700, 4450, -41456, -356232, 3893280, 41666730, ...}.

Examples

			Triangle begins as:
  1;
  1,     1;
  1,     2,     1;
  1,   -11,   -11,      1;
  1,   -23,   -36,    -23,      1;
  1,   187,   162,    162,    187,      1;
  1,   781,   966,    954,    966,    781,      1;
  1, -7495, -6716,  -6518,  -6518,  -6716,  -7495,    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:
A176307 := 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) = 2 - A176306(n,k).