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

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, 273, 2, 3, 10, 24, 65, 168, 442, 714, 2, 3, 10, 24, 65, 168, 442, 1155, 1870, 2, 3, 10, 24, 65, 168, 442, 1155, 3026, 4895, 2, 3, 10, 24, 65, 168, 442, 1155, 3026, 7920, 12816

Offset: 1

Author

Werner Schulte, Nov 21 2024

Keywords

• nonn
• easy
• tabl
• frac

Comments

The harmonic triangle uses the terms of this sequence as denominators, numerators = 1.

The inverse of the harmonic triangle has entries -(Fibonacci(k+1))^2 for 1<=k

Row sums of the harmonic triangle are 1.

Conjecture: Alt. row sums of the harmonic triangle are Fibonacci(n-2) / Fibonacci(n+1), where Fibonacci(-1) = 1.

Examples

			Triangle T(n, k) for 1 <= k <= n starts:
n\ k :  1  2   3   4   5    6    7     8     9    10     11
===========================================================
   1 :  1
   2 :  2  2
   3 :  2  3   6
   4 :  2  3  10  15
   5 :  2  3  10  24  40
   6 :  2  3  10  24  65  104
   7 :  2  3  10  24  65  168  273
   8 :  2  3  10  24  65  168  442   714
   9 :  2  3  10  24  65  168  442  1155  1870
  10 :  2  3  10  24  65  168  442  1155  3026  4895
  11 :  2  3  10  24  65  168  442  1155  3026  7920  12816
  etc.
		

Crossrefs

Cf. A000045, A110034, A110035, A001654 (main diagonal), A059929 (subdiagonals).

Programs

PARI

T(n,k)=if(k==n,Fibonacci(n)*Fibonacci(n+1),Fibonacci(k)*Fibonacci(k+2))

Formula

T(n, k) = Fibonacci(n) * Fibonacci(n+1) if k = n, and Fibonacci(k) * Fibonacci(k+2) if 1 <= k < n.

Row sums are A110035(n) - 1 = -A110034(n+1).

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)).

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