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%I #5 Dec 08 2024 17:12:17
%S 1,2,2,2,3,6,2,3,10,15,2,3,10,24,40,2,3,10,24,65,104,2,3,10,24,65,168,
%T 273,2,3,10,24,65,168,442,714,2,3,10,24,65,168,442,1155,1870,2,3,10,
%U 24,65,168,442,1155,3026,4895,2,3,10,24,65,168,442,1155,3026,7920,12816
%N Denominators in a harmonic triangle, based on products of Fibonacci numbers.
%C The harmonic triangle uses the terms of this sequence as denominators, numerators = 1.
%C The inverse of the harmonic triangle has entries -(Fibonacci(k+1))^2 for 1<=k<n (subdiagonals) and Fibonacci(n) * Fibonacci(n+1) (main diagonal).
%C Row sums of the harmonic triangle are 1.
%C Conjecture: Alt. row sums of the harmonic triangle are Fibonacci(n-2) / Fibonacci(n+1), where Fibonacci(-1) = 1.
%F T(n, k) = Fibonacci(n) * Fibonacci(n+1) if k = n, and Fibonacci(k) * Fibonacci(k+2) if 1 <= k < n.
%F Row sums are A110035(n) - 1 = -A110034(n+1).
%F G.f.: A(t, x) = x*t*(1 + t - x*t^2) / ((1 - t) * (1 + x*t) * (1 - 3*x*t + x^2*t^2)).
%e Triangle T(n, k) for 1 <= k <= n starts:
%e n\ k : 1 2 3 4 5 6 7 8 9 10 11
%e ===========================================================
%e 1 : 1
%e 2 : 2 2
%e 3 : 2 3 6
%e 4 : 2 3 10 15
%e 5 : 2 3 10 24 40
%e 6 : 2 3 10 24 65 104
%e 7 : 2 3 10 24 65 168 273
%e 8 : 2 3 10 24 65 168 442 714
%e 9 : 2 3 10 24 65 168 442 1155 1870
%e 10 : 2 3 10 24 65 168 442 1155 3026 4895
%e 11 : 2 3 10 24 65 168 442 1155 3026 7920 12816
%e etc.
%o (PARI) T(n,k)=if(k==n,Fibonacci(n)*Fibonacci(n+1),Fibonacci(k)*Fibonacci(k+2))
%Y Cf. A000045, A110034, A110035, A001654 (main diagonal), A059929 (subdiagonals).
%K nonn,easy,tabl,frac,new
%O 1,2
%A _Werner Schulte_, Nov 21 2024