A176340 Triangle T(n,k) = 1 - A176338(k) - A176338(n-k) + A176338(n) read by rows.

1, 1, 1, 1, -8, 1, 1, 190, 190, 1, 1, -14822, -14624, -14822, 1, 1, 3557278, 3542464, 3542464, 3557278, 1, 1, -2582583830, -2579026544, -2579041556, -2579026544, -2582583830, 1, 1, 5640363084718, 5637780500896, 5637784057984, 5637784057984, 5637780500896, 5640363084718, 1

Offset

0,5

Comments

Row sums are: {1, 2, -6, 382, -44266, 14199486, -12902262302, 33831855287198,

-258898313695820850, 5823405140242006622494, -386839522966544578870468774, ...}.

Links

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

Formula

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

Example

Triangle starts as:
  1;
  1,       1;
  1,      -8,       1;
  1,     190,     190,       1;
  1,  -14822,  -14624,  -14822,       1;
  1, 3557278, 3542464, 3542464, 3557278, 1;

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, 3], {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, 3), ", "))) \\ 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, 3) : 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, 3) 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, 3) ))); # G. C. Greubel, Dec 07 2019

Crossrefs

Cf. A176337, A176338, A176339.

Sequence in context: A015121 A156766 A178046 * A111835 A254460 A198830

Adjacent sequences: A176337 A176338 A176339 * A176341 A176342 A176343

Keyword

sign,tabl

Author

Roger L. Bagula, Apr 15 2010

Extensions

Edited by G. C. Greubel, Dec 07 2019

Status

approved