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

%I A176340 #13 Sep 08 2022 08:45:52
%S A176340 1,1,1,1,-8,1,1,190,190,1,1,-14822,-14624,-14822,1,1,3557278,3542464,
%T A176340 3542464,3557278,1,1,-2582583830,-2579026544,-2579041556,-2579026544,
%U A176340 -2582583830,1,1,5640363084718,5637780500896,5637784057984,5637784057984,5637780500896,5640363084718,1
%N A176340 Triangle T(n,k) = 1 - A176338(k) - A176338(n-k) + A176338(n) read by rows.
%C A176340 Row sums are: {1, 2, -6, 382, -44266, 14199486, -12902262302, 33831855287198,
%C A176340 -258898313695820850, 5823405140242006622494, -386839522966544578870468774, ...}.
%H A176340 G. C. Greubel, <a href="/A176340/b176340.txt">Rows n = 0..25 of triangle, flattened</a>
%F A176340 T(n,k) = T(n,n-k).
%e A176340 Triangle starts as:
%e A176340   1;
%e A176340   1,       1;
%e A176340   1,      -8,       1;
%e A176340   1,     190,     190,       1;
%e A176340   1,  -14822,  -14624,  -14822,       1;
%e A176340   1, 3557278, 3542464, 3542464, 3557278, 1;
%t A176340 b[n_, q_]:= b[n, q]= If[n==0, 0, (1-q^n)*b[n-1, q] +1];
%t A176340 T[n_,k_,q_]:= 1 + b[n,q] -b[n-k,q] - b[k,q];
%t A176340 Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Dec 07 2019 *)
%o A176340 (PARI) b(n,q) = if(n==0, 0, 1 + (1-q^n)*b(n-1,q) );
%o A176340 T(n,k,q) = 1 + b(n,q) - b(n-k,q) - b(k,q);
%o A176340 for(n=0,10, for(k=0,n, print1(T(n,k,3), ", "))) \\ _G. C. Greubel_, Dec 07 2019
%o A176340 (Magma)
%o A176340 function b(n,q)
%o A176340   if n eq 0 then return 0;
%o A176340   else return 1 - (q^n-1)*b(n-1,q);
%o A176340   end if; return b; end function;
%o A176340 function T(n,k,q) return 1 + b(n,q) - b(n-k,q) - b(k,q); end function;
%o A176340 [ T(n,k,3) : k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 07 2019
%o A176340 (Sage)
%o A176340 @CachedFunction
%o A176340 def b(n, q):
%o A176340     if (n==0): return 0
%o A176340     else: return 1 - (q^n-1)*b(n-1,q)
%o A176340 def T(n,k,q): return 1 + b(n,q) - b(n-k,q) - b(k,q)
%o A176340 [[T(n,k,3) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 07 2019
%o A176340 (GAP)
%o A176340 b:= function(n,q)
%o A176340     if n=0 then return 0;
%o A176340     else return 1 - (q^n-1)*b(n-1,q);
%o A176340     fi; end;
%o A176340 T:= function(n,k,q) return 1 + b(n,q) - b(n-k,q) - b(k,q); end;
%o A176340 Flat(List([0..10], n-> List([0..n], k-> T(n,k,3) ))); # _G. C. Greubel_, Dec 07 2019
%Y A176340 Cf. A176337, A176338, A176339.
%K A176340 sign,tabl
%O A176340 0,5
%A A176340 _Roger L. Bagula_, Apr 15 2010
%E A176340 Edited by _G. C. Greubel_, Dec 07 2019