A176339 Triangle T(n,k) = 1 - A176337(k) - A176337(n-k) + A176337(n) read by rows.

1, 1, 1, 1, -3, 1, 1, 17, 17, 1, 1, -239, -219, -239, 1, 1, 7169, 6933, 6933, 7169, 1, 1, -444479, -437307, -437563, -437307, -444479, 1, 1, 56004353, 55559877, 55567029, 55567029, 55559877, 56004353, 1, 1, -14225105663, -14169101307, -14169545803, -14169538395, -14169545803, -14169101307, -14225105663, 1

Offset: 0

Author

Roger L. Bagula, Apr 15 2010

Keywords

• sign
• tabl

Comments

Row sums are {1, 2, -1, 36, -695, 28206, -2201133, 334262520, -99297043939, 57953303599938, -66678973493759897, ...}.

Examples

			Triangle begins as:
  1;
  1,       1;
  1,      -3,       1;
  1,      17,      17,       1;
  1,    -239,    -219,    -239,       1;
  1,    7169,    6933,    6933,    7169,       1;
  1, -444479, -437307, -437563, -437307, -444479, 1;
		

Links

• G. C. Greubel, Rows n = 0..25 of triangle, flattened

Crossrefs

Cf. A176337, A176338, A176340.

Programs

GAP

b:= function(n,q)
    if n=0 then return 0;
    else return 1 - (q^n-1)*b(n-1,q);
    fi; end;
T:= function(n,k,q) return 1 + b(n,q) - b(n-k,q) - b(k,q); end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k,2) ))); # G. C. Greubel, Dec 07 2019

Magma

function b(n,q)
  if n eq 0 then return 0;
  else return 1 - (q^n-1)*b(n-1,q);
  end if; return b; end function;
function T(n,k,q) return 1 + b(n,q) - b(n-k,q) - b(k,q); end function;
[ T(n,k,2) : k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 07 2019

Maple

A176339 := proc(n,m)
    1-A176337(m)-A176337(n-m)+A176337(n) ;
end proc: # R. J. Mathar, May 04 2013

Mathematica

b[n_, q_]:= b[n, q]= If[n==0, 0, (1-q^n)*b[n-1, q] +1];
T[n_,k_,q_]:= 1 + b[n,q] -b[n-k,q] - b[k,q];
Table[T[n,k,2], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Dec 07 2019 *)

PARI

b(n,q) = if(n==0, 0, 1 + (1-q^n)*b(n-1,q) );
T(n,k,q) = 1 + b(n,q) - b(n-k,q) - b(k,q);
for(n=0,10, for(k=0,n, print1(T(n,k,2), ", "))) \\ G. C. Greubel, Dec 07 2019

Sage

@CachedFunction
def b(n, q):
    if (n==0): return 0
    else: return 1 - (q^n-1)*b(n-1,q)
def T(n,k,q): return 1 + b(n,q) - b(n-k,q) - b(k,q)
[[T(n,k,2) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 07 2019

Formula

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