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%I #12 Apr 28 2021 02:09:23
%S 1,1,1,1,1,1,1,2,2,1,1,2,4,2,1,1,3,6,6,3,1,1,4,12,12,12,4,1,1,4,16,24,
%T 24,16,4,1,1,4,16,32,48,32,16,4,1,1,5,20,40,80,80,40,20,5,1,1,6,30,60,
%U 120,160,120,60,30,6,1
%N Triangle T(n, k) = round( A172452(n)/(A172452(k)*A172452(n-k)) ), read by rows.
%C The original definition of this sequence did not produce an integer valued triangular sequence. The application of the "round" function was the method chosen to formulate an integer sequence. - _G. C. Greubel_, Apr 27 2021
%H G. C. Greubel, <a href="/A172453/b172453.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = round( A172452(n)/(A172452(k)*A172452(n-k)) ).
%F T(n, k) = round( c(n)/(c(k)*c(n-k)) ) where c(n) = Product_{j=1..n} A004001(j) with c(0) = 1.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 2, 2, 1;
%e 1, 2, 4, 2, 1;
%e 1, 3, 6, 6, 3, 1;
%e 1, 4, 12, 12, 12, 4, 1;
%e 1, 4, 16, 24, 24, 16, 4, 1;
%e 1, 4, 16, 32, 48, 32, 16, 4, 1;
%e 1, 5, 20, 40, 80, 80, 40, 20, 5, 1;
%e 1, 6, 30, 60, 120, 160, 120, 60, 30, 6, 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_]:= Round[c[n]/(c[k]*c[n-k])];
%t Table[T[n, k], {n,0,10}, {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 round(c(n)/(c(k)*c(n-k)))
%o [[T(n,k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Apr 27 2021
%Y Cf. A004001, A172452.
%K nonn,tabl,easy,less
%O 0,8
%A _Roger L. Bagula_, Feb 03 2010
%E Definition changed to give integral terms and edited by _G. C. Greubel_, Apr 27 2021
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