cp's OEIS Frontend

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

%I A377278 #8 Nov 15 2024 23:33:31
%S A377278 1,3,3,3,21,7,3,21,105,15,3,21,105,465,31,3,21,105,465,1953,63,3,21,
%T A377278 105,465,1953,8001,127,3,21,105,465,1953,8001,32385,255,3,21,105,465,
%U A377278 1953,8001,32385,130305,511,3,21,105,465,1953,8001,32385,130305,522753,1023
%N A377278 Denominators in a harmonic triangle; q-analog of A126615, here q = 2.
%C A377278 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).
%C A377278 Conjecture: Row sums of the harmonic triangle are A204243(n) / A005329(n).
%F A377278 T(n, k) = (2^k - 1) * (2^(k+1) - 1) for 1 <= k < n; T(n, n) = 2^n - 1.
%F A377278 Sum_{k=1..n} 2^(k-1) / T(n, k) = 1.
%F A377278 Product_{k=1..n} T(n, k)^((-1)^k) = 1.
%F A377278 Row sums are n + 4 * (2^n - 1) * (2^(n-1) - 1) / 3 = n + 4 * A006095(n).
%F A377278 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
%e A377278 Triangle T(n, k) for 1 <= k <= n starts:
%e A377278 n\ k :  1   2    3    4     5     6      7       8       9    10
%e A377278 ================================================================
%e A377278    1 :  1
%e A377278    2 :  3   3
%e A377278    3 :  3  21    7
%e A377278    4 :  3  21  105   15
%e A377278    5 :  3  21  105  465    31
%e A377278    6 :  3  21  105  465  1953    63
%e A377278    7 :  3  21  105  465  1953  8001    127
%e A377278    8 :  3  21  105  465  1953  8001  32385     255
%e A377278    9 :  3  21  105  465  1953  8001  32385  130305     511
%e A377278   10 :  3  21  105  465  1953  8001  32385  130305  522753  1023
%e A377278   etc.
%e A377278 The harmonic triangle starts:
%e A377278   [1]  1/1
%e A377278   [2]  1/3   1/3
%e A377278   [3]  1/3  1/21    1/7
%e A377278   [4]  1/3  1/21  1/105   1/15
%e A377278   [5]  1/3  1/21  1/105  1/465    1/31
%e A377278   [6]  1/3  1/21  1/105  1/465  1/1953  1/63
%e A377278   etc.
%e A377278 The inverse of the harmonic triangle starts:
%e A377278   [1]    1
%e A377278   [2]   -1   3
%e A377278   [3]   -2  -1   7
%e A377278   [4]   -4  -2  -1  15
%e A377278   [5]   -8  -4  -2  -1  31
%e A377278   [6]  -16  -8  -4  -2  -1  63
%e A377278   etc.
%o A377278 (PARI)  T(n,k)=if(k<n,(2^k-1)*(2^(k+1)-1),2^n-1)
%Y A377278 Cf. A000225, A005329, A006095, A126615, A204243.
%K A377278 nonn,easy,tabl,frac
%O A377278 1,2
%A A377278 _Werner Schulte_, Oct 22 2024