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

%I A176339 #15 Sep 08 2022 08:45:52
%S A176339 1,1,1,1,-3,1,1,17,17,1,1,-239,-219,-239,1,1,7169,6933,6933,7169,1,1,
%T A176339 -444479,-437307,-437563,-437307,-444479,1,1,56004353,55559877,
%U A176339 55567029,55567029,55559877,56004353,1,1,-14225105663,-14169101307,-14169545803,-14169538395,-14169545803,-14169101307,-14225105663,1
%N A176339 Triangle T(n,k) = 1 - A176337(k) - A176337(n-k) + A176337(n) read by rows.
%C A176339 Row sums are {1, 2, -1, 36, -695, 28206, -2201133, 334262520, -99297043939, 57953303599938, -66678973493759897, ...}.
%H A176339 G. C. Greubel, <a href="/A176339/b176339.txt">Rows n = 0..25 of triangle, flattened</a>
%F A176339 T(n,k) = T(n,n-k).
%e A176339 Triangle begins as:
%e A176339   1;
%e A176339   1,       1;
%e A176339   1,      -3,       1;
%e A176339   1,      17,      17,       1;
%e A176339   1,    -239,    -219,    -239,       1;
%e A176339   1,    7169,    6933,    6933,    7169,       1;
%e A176339   1, -444479, -437307, -437563, -437307, -444479, 1;
%p A176339 A176339 := proc(n,m)
%p A176339     1-A176337(m)-A176337(n-m)+A176337(n) ;
%p A176339 end proc: # _R. J. Mathar_, May 04 2013
%t A176339 b[n_, q_]:= b[n, q]= If[n==0, 0, (1-q^n)*b[n-1, q] +1];
%t A176339 T[n_,k_,q_]:= 1 + b[n,q] -b[n-k,q] - b[k,q];
%t A176339 Table[T[n,k,2], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Dec 07 2019 *)
%o A176339 (PARI) b(n,q) = if(n==0, 0, 1 + (1-q^n)*b(n-1,q) );
%o A176339 T(n,k,q) = 1 + b(n,q) - b(n-k,q) - b(k,q);
%o A176339 for(n=0,10, for(k=0,n, print1(T(n,k,2), ", "))) \\ _G. C. Greubel_, Dec 07 2019
%o A176339 (Magma) function b(n,q)
%o A176339   if n eq 0 then return 0;
%o A176339   else return 1 - (q^n-1)*b(n-1,q);
%o A176339   end if; return b; end function;
%o A176339 function T(n,k,q) return 1 + b(n,q) - b(n-k,q) - b(k,q); end function;
%o A176339 [ T(n,k,2) : k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 07 2019
%o A176339 (Sage)
%o A176339 @CachedFunction
%o A176339 def b(n, q):
%o A176339     if (n==0): return 0
%o A176339     else: return 1 - (q^n-1)*b(n-1,q)
%o A176339 def T(n,k,q): return 1 + b(n,q) - b(n-k,q) - b(k,q)
%o A176339 [[T(n,k,2) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 07 2019
%o A176339 (GAP)
%o A176339 b:= function(n,q)
%o A176339     if n=0 then return 0;
%o A176339     else return 1 - (q^n-1)*b(n-1,q);
%o A176339     fi; end;
%o A176339 T:= function(n,k,q) return 1 + b(n,q) - b(n-k,q) - b(k,q); end;
%o A176339 Flat(List([0..10], n-> List([0..n], k-> T(n,k,2) ))); # _G. C. Greubel_, Dec 07 2019
%Y A176339 Cf. A176337, A176338, A176340.
%K A176339 sign,tabl
%O A176339 0,5
%A A176339 _Roger L. Bagula_, Apr 15 2010