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A378277 Denominators in a harmonic triangle, based on products of Fibonacci numbers.

%I A378277 #5 Dec 08 2024 17:12:17
%S A378277 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 A378277 273,2,3,10,24,65,168,442,714,2,3,10,24,65,168,442,1155,1870,2,3,10,
%U A378277 24,65,168,442,1155,3026,4895,2,3,10,24,65,168,442,1155,3026,7920,12816
%N A378277 Denominators in a harmonic triangle, based on products of Fibonacci numbers.
%C A378277 The harmonic triangle uses the terms of this sequence as denominators, numerators = 1.
%C A378277 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 A378277 Row sums of the harmonic triangle are 1.
%C A378277 Conjecture: Alt. row sums of the harmonic triangle are Fibonacci(n-2) / Fibonacci(n+1), where Fibonacci(-1) = 1.
%F A378277 T(n, k) = Fibonacci(n) * Fibonacci(n+1) if k = n, and Fibonacci(k) * Fibonacci(k+2) if 1 <= k < n.
%F A378277 Row sums are A110035(n) - 1 = -A110034(n+1).
%F A378277 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 A378277 Triangle T(n, k) for 1 <= k <= n starts:
%e A378277 n\ k :  1  2   3   4   5    6    7     8     9    10     11
%e A378277 ===========================================================
%e A378277    1 :  1
%e A378277    2 :  2  2
%e A378277    3 :  2  3   6
%e A378277    4 :  2  3  10  15
%e A378277    5 :  2  3  10  24  40
%e A378277    6 :  2  3  10  24  65  104
%e A378277    7 :  2  3  10  24  65  168  273
%e A378277    8 :  2  3  10  24  65  168  442   714
%e A378277    9 :  2  3  10  24  65  168  442  1155  1870
%e A378277   10 :  2  3  10  24  65  168  442  1155  3026  4895
%e A378277   11 :  2  3  10  24  65  168  442  1155  3026  7920  12816
%e A378277   etc.
%o A378277 (PARI) T(n,k)=if(k==n,Fibonacci(n)*Fibonacci(n+1),Fibonacci(k)*Fibonacci(k+2))
%Y A378277 Cf. A000045, A110034, A110035, A001654 (main diagonal), A059929 (subdiagonals).
%K A378277 nonn,easy,tabl,frac
%O A378277 1,2
%A A378277 _Werner Schulte_, Nov 21 2024