A377278 Denominators in a harmonic triangle; q-analog of A126615, here q = 2.

1, 3, 3, 3, 21, 7, 3, 21, 105, 15, 3, 21, 105, 465, 31, 3, 21, 105, 465, 1953, 63, 3, 21, 105, 465, 1953, 8001, 127, 3, 21, 105, 465, 1953, 8001, 32385, 255, 3, 21, 105, 465, 1953, 8001, 32385, 130305, 511, 3, 21, 105, 465, 1953, 8001, 32385, 130305, 522753, 1023

Offset: 1

Author

Werner Schulte, Oct 22 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 -2^(n-k-1) for 1<=k

Conjecture: Row sums of the harmonic triangle are A204243(n) / A005329(n).

Examples

			Triangle T(n, k) for 1 <= k <= n starts:
n\ k :  1   2    3    4     5     6      7       8       9    10
================================================================
   1 :  1
   2 :  3   3
   3 :  3  21    7
   4 :  3  21  105   15
   5 :  3  21  105  465    31
   6 :  3  21  105  465  1953    63
   7 :  3  21  105  465  1953  8001    127
   8 :  3  21  105  465  1953  8001  32385     255
   9 :  3  21  105  465  1953  8001  32385  130305     511
  10 :  3  21  105  465  1953  8001  32385  130305  522753  1023
  etc.
The harmonic triangle starts:
  [1]  1/1
  [2]  1/3   1/3
  [3]  1/3  1/21    1/7
  [4]  1/3  1/21  1/105   1/15
  [5]  1/3  1/21  1/105  1/465    1/31
  [6]  1/3  1/21  1/105  1/465  1/1953  1/63
  etc.
The inverse of the harmonic triangle starts:
  [1]    1
  [2]   -1   3
  [3]   -2  -1   7
  [4]   -4  -2  -1  15
  [5]   -8  -4  -2  -1  31
  [6]  -16  -8  -4  -2  -1  63
  etc.
		

Crossrefs

Cf. A000225, A005329, A006095, A126615, A204243.

Programs

PARI

T(n,k)=if(k
				

Formula

T(n, k) = (2^k - 1) * (2^(k+1) - 1) for 1 <= k < n; T(n, n) = 2^n - 1.

Sum_{k=1..n} 2^(k-1) / T(n, k) = 1.

Product_{k=1..n} T(n, k)^((-1)^k) = 1.

Row sums are n + 4 * (2^n - 1) * (2^(n-1) - 1) / 3 = n + 4 * A006095(n).

G.f.: x*y*(1 + 2*x - 4*x*y + 4*x^2*y)/((1 - x)*(1 - x*y)(1 - 2*x*y)*(1 - 4*x*y)). - Stefano Spezia, Oct 23 2024

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