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%I #8 Apr 27 2021 06:06:58
%S -1,-1,-1,-1,-1,-1,-1,0,0,-1,-1,1,2,1,-1,-1,7,9,9,7,-1,-1,35,43,44,43,
%T 35,-1,-1,143,179,186,186,179,143,-1,-1,575,719,754,760,754,719,575,
%U -1,-1,3071,3647,3790,3824,3824,3790,3647,3071,-1,-1,19199,22271,22846,22988,23016,22988,22846,22271,19199,-1
%N Triangle T(n, k) = A172452(n) - A172452(k) - A172452(n-k), read by rows.
%C Row sums are: {-1, -2, -3, -2, 2, 30, 198, 1014, 4854, 28662, 197622, ...}.
%H G. C. Greubel, <a href="/A172970/b172970.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = A172452(n) - A172452(k) - A172452(n-k).
%F T(n, k) = c(n) - c(k) - c(n-k) where c(n) = Product_{j=1..n} A004001(j).
%e Triangle begins as:
%e -1;
%e -1, -1;
%e -1, -1, -1;
%e -1, 0, 0, -1;
%e -1, 1, 2, 1, -1;
%e -1, 7, 9, 9, 7, -1;
%e -1, 35, 43, 44, 43, 35, -1;
%e -1, 143, 179, 186, 186, 179, 143, -1;
%e -1, 575, 719, 754, 760, 754, 719, 575, -1;
%e -1, 3071, 3647, 3790, 3824, 3824, 3790, 3647, 3071, -1;
%e -1, 19199, 22271, 22846, 22988, 23016, 22988, 22846, 22271, 19199, -1;
%t f[n_]:= f[n]= If[n<3, Fibonacci[n], f[f[n-1]] + f[n-f[n-1]]]; (* f=A004001 *)
%t c[n_]:= Product[f[j], {j,n}]; (* c=A172452 *)
%t T[n_, k_]:= c[n] - c[k] - c[n-k];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Apr 27 2021 *)
%o (Sage)
%o @CachedFunction
%o def b(n): return fibonacci(n) if (n<3) else b(b(n-1)) + b(n-b(n-1)) # b=A004001
%o def c(n): return product(b(j) for j in (1..n)) # c=A172452
%o def T(n,k): return c(n) - c(k) - c(n-k)
%o [[T(n,k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Apr 27 2021
%Y Cf. A004001, A172452, A172453.
%K sign,tabl
%O 0,13
%A _Roger L. Bagula_, Feb 06 2010
%E Edited by _G. C. Greubel_, Apr 27 2021
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