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A176344 T(n,k) = 1 + A176343(n) - A176343(k) - A176343(n-k), triangle read by rows (n >= 0, 0 <= k <= n).

%I A176344 #15 Sep 08 2022 08:45:52
%S A176344 1,1,1,1,1,1,1,3,3,1,1,11,13,11,1,1,65,75,75,65,1,1,568,632,640,632,
%T A176344 568,1,1,7789,8356,8418,8418,8356,7789,1,1,168761,176549,177114,
%U A176344 177168,177114,176549,168761,1,1,5847568,6016328,6024114,6024671,6024671,6024114,6016328,5847568,1
%N A176344 T(n,k) = 1 + A176343(n) - A176343(k) - A176343(n-k), triangle read by rows (n >= 0, 0 <= k <= n).
%H A176344 G. C. Greubel, <a href="/A176344/b176344.txt">Rows n = 0..75 of triangle, flattened</a>
%e A176344 Triangle begins:
%e A176344   1;
%e A176344   1,      1;
%e A176344   1,      1,      1;
%e A176344   1,      3,      3,      1;
%e A176344   1,     11,     13,     11,      1;
%e A176344   1,     65,     75,     75,     65,      1;
%e A176344   1,    568,    632,    640,    632,    568,      1;
%e A176344   1,   7789,   8356,   8418,   8418,   8356,   7789,      1;
%e A176344   1, 168761, 176549, 177114, 177168, 177114, 176549, 168761, 1;
%e A176344   ...
%p A176344 with(combinat);
%p A176344 b:= proc(n) option remember;
%p A176344    if n = 0 then 0    else 1+fibonacci(n)*b(n-1)
%p A176344    fi; end proc;
%p A176344 T:= proc (n, k) 1+b(n)-b(n-k)-b(k) end proc;
%p A176344 seq(seq(T(n, k), k = 0..n), n = 0..10); # _G. C. Greubel_, Dec 08 2019
%t A176344 b[n_]:= b[n]= If[n==0, 0, Fibonacci[n]*b[n-1] + 1]; (* A176343 *)
%t A176344 T[n_, k_]:= T[n, k] = 1 + a[n] - a[n-k] - a[k];
%t A176344 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Dec 08 2019 *)
%o A176344 (Maxima) (a[0] : 0, a[n] := fib(n)*a[n-1] + 1, T(n, m) := 1 + a[n] - a[m] - a[n-m])$ create_list(T(n, m), n, 0, 10, m, 0, n); /* _Franck Maminirina Ramaharo_, Nov 25 2018 */
%o A176344 (PARI) b(n) = if(n==0, 0, 1 + fibonacci(n)*b(n-1) );
%o A176344 T(n,k) = 1 + b(n) - b(n-k) - b(k);
%o A176344 for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Dec 07 2019
%o A176344 (Magma)
%o A176344 function b(n)
%o A176344   if n eq 0 then return 0;
%o A176344   else return 1 + Fibonacci(n)*b(n-1);
%o A176344   end if; return b; end function;
%o A176344 function T(n,k) return 1 + b(n) - b(n-k) - b(k); end function; [ T(n,k) : k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 07 2019
%o A176344 (Sage)
%o A176344 def b(n):
%o A176344     if (n==0): return 0
%o A176344     else: return 1 + fibonacci(n)*b(n-1)
%o A176344 def T(n,k): return 1 + b(n) - b(n-k) - b(k)
%o A176344 [[T(n,k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 07 2019
%o A176344 (GAP)
%o A176344 b:= function(n)
%o A176344     if n=0 then return 0;
%o A176344     else return 1 + Fibonacci(n)*b(n-1);
%o A176344     fi; end;
%o A176344 T:= function(n,k) return 1 + b(n) - b(n-k) - b(k); end;
%o A176344 Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Dec 07 2019
%Y A176344 Cf. A156070, A156072, A176305, A176306, A176307, A176625, A176339.
%K A176344 nonn,tabl,easy
%O A176344 0,8
%A A176344 _Roger L. Bagula_, Apr 15 2010
%E A176344 Edited and name clarified by _Franck Maminirina Ramaharo_, Nov 25 2018