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A377278
Denominators in a harmonic triangle; q-analog of A126615, here q = 2.
0
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
1,2
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<n (subdiagonals) and 2^n - 1 (main diagonal).
Conjecture: Row sums of the harmonic triangle are A204243(n) / A005329(n).
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
EXAMPLE
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.
PROG
(PARI) T(n, k)=if(k<n, (2^k-1)*(2^(k+1)-1), 2^n-1)
CROSSREFS
KEYWORD
nonn,easy,tabl,frac
AUTHOR
Werner Schulte, Oct 22 2024
STATUS
approved