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

0,5

Comments

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

Links

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

Formula

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

Example

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;

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 *)

Prog

(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
(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
(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
(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

Crossrefs

Cf. A176337, A176338, A176340.

Sequence in context: A322790 A333560 A176293 * A121412 A212855 A016561

Adjacent sequences: A176336 A176337 A176338 * A176340 A176341 A176342

Keyword

sign,tabl

Author

Roger L. Bagula, Apr 15 2010

Status

approved