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A120077 Denominators of row sums of rational triangle A120072/A120073.

%I A120077 #18 Apr 26 2023 08:54:49
%S A120077 4,36,144,3600,3600,176400,705600,6350400,1270080,153679680,153679680,
%T A120077 25971865920,25971865920,129859329600,519437318400,150117385017600,
%U A120077 150117385017600,54192375991353600,2167695039654144,1548353599752960,221193371393280,117011293467045120
%N A120077 Denominators of row sums of rational triangle A120072/A120073.
%C A120077 The first 19 terms coincide with A007407(n), for n>=2. However a(20) = 2167695039654144 and A007407(20) = 10838475198270720 = 5*a(20). Also a(21) = 1548353599752960 and A007407(21) = 221193371393280 = a(21)/7. From n = 22 up to at least n = 100 (checked) both sequences coincide again.
%C A120077 See the W. Lang link under A120072 for more details.
%C A120077 The corresponding numerators are given by A120076.
%C A120077 The n for which a(n) differs from A007407(n) are given by A309829. - _Jeppe Stig Nielsen_, Aug 18 2019
%H A120077 Jeppe Stig Nielsen, <a href="/A120077/b120077.txt">Table of n, a(n) for n = 2..1150</a>
%F A120077 a(n) = denominator(r(m)), with the rationals r(m) = Sum_{n=1..m-1} A120072(m,n)/A120073(m,n), m >= 2.
%F A120077 The rationals are r(m) = Zeta(2; m-1) - (m-1)/m^2, m>=2, with the partial sums Zeta(2; n) = Sum_{k=1..n} 1/k^2. See the W. Lang link under A103345.
%F A120077 O.g.f. for the rationals r(m), m>=2: log(1-x) + polylog(2,x)/(1-x).
%e A120077 The rationals A120076(m)/a(m), m>=2, begin with (3/4, 37/36, 169/144, 4549/3600, 4769/3600, ... ).
%t A120077 Table[Denominator[HarmonicNumber[n,2] -1/n], {n,2,40}] (* _G. C. Greubel_, Apr 25 2023 *)
%o A120077 (PARI) a(n) = denominator(sum(j=1,n-1,1/j^2-1/n^2)) \\ _Jeppe Stig Nielsen_, Aug 18 2019
%o A120077 (PARI) a(n) = denominator(sum(j=1,n,1/j^2) - 1/n) \\ _Jeppe Stig Nielsen_, Aug 18 2019
%o A120077 (Magma)
%o A120077 A120077:= func< n | Denominator( (&+[1/k^2: k in [1..n]]) -1/n) >;
%o A120077 [A120077(n): n in [2..30]]; // _G. C. Greubel_, Apr 25 2023
%o A120077 (SageMath)
%o A120077 def A120077(n): return denominator(harmonic_number(n,2) - 1/n)
%o A120077 [A120077(n) for n in range(2,31)] # _G. C. Greubel_, Apr 25 2023
%Y A120077 Cf. A007407, A120070, A120072, A120073, A120074, A120075, A120076, A309829.
%K A120077 nonn,easy,frac
%O A120077 2,1
%A A120077 _Wolfdieter Lang_, Jul 20 2006
%E A120077 a(21)-a(23) from _Jeppe Stig Nielsen_, Aug 18 2019