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A230328 Denominator of n(n+3)/(4(n+1)(n+2)) = Sum_{k = 1..n} 1/(k(k+1)(k+2)).

%I A230328 #25 Jan 08 2025 10:45:37
%S A230328 1,6,24,40,30,21,112,144,45,110,264,312,182,105,480,544,153,342,760,
%T A230328 840,462,253,1104,1200,325,702,1512,1624,870,465,1984,2112,561,1190,
%U A230328 2520,2664,1406,741,3120,3280,861,1806,3784,3960,2070,1081,4512
%N A230328 Denominator of n(n+3)/(4(n+1)(n+2)) = Sum_{k = 1..n} 1/(k(k+1)(k+2)).
%H A230328 Jean-François Alcover, <a href="/A230328/b230328.txt">Table of n, a(n) for n = 0..1000</a>
%H A230328 Peter Bala, <a href="/A306367/a306367.pdf">A note on the sequence of numerators of a rational function</a>
%H A230328 Wikipedia, <a href="https://en.wikipedia.org/wiki/Quasi-polynomial">Quasi-polynomial</a>
%F A230328 G.f.: -(x^18 +3*x^17 +12*x^16 -6*x^15 +9*x^14 +91*x^12 -138*x^11 +183*x^10 -134*x^9 +183*x^8 -138*x^7 +91*x^6 +9*x^4 -6*x^3 +12*x^2 +3*x +1) / ((x -1)^3*(x^2 +1)^3*(x^4 +1)^3). - _Colin Barker_, Oct 09 2014 [Confirmed by _Peter Bala_, Feb 27 2019]
%F A230328 From _Peter Bala_, Feb 26 2019: (Start)
%F A230328 a(n) = 4*(n + 1)*(n + 2)/gcd(4*(n + 1)*(n + 2), n*(n + 3)).
%F A230328 a(n) is quasi-polynomial in n; a(n) = (n + 1)*(n + 2)/2 when n == 0, 5 (mod 8); a(n) = (n + 1)*(n + 2) when n == 1, 4 (mod 8); a(n) = 2*(n + 1)*(n + 2) when n == 2, 3, 6, 7 (mod 8). (End)
%p A230328 seq( 4*(n + 1)*(n + 2)/igcd(4*(n + 1)*(n + 2), n*(n + 3)), n = 0..100); # _Peter Bala_, Feb 26 2019
%t A230328 Table[Denominator[n*(n+3)/(4*(n+1)*(n+2))], {n, 0, 100}]
%o A230328 (PARI) vector(100, n, denominator((n-1)*(n+2)/(4*n*(n+1)))) \\ _Colin Barker_, Oct 09 2014
%Y A230328 Cf. A125650 (numerators).
%K A230328 nonn,frac,easy
%O A230328 0,2
%A A230328 _Jean-François Alcover_, Oct 16 2013