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

1, 6, 24, 40, 30, 21, 112, 144, 45, 110, 264, 312, 182, 105, 480, 544, 153, 342, 760, 840, 462, 253, 1104, 1200, 325, 702, 1512, 1624, 870, 465, 1984, 2112, 561, 1190, 2520, 2664, 1406, 741, 3120, 3280, 861, 1806, 3784, 3960, 2070, 1081, 4512

Offset: 0

Author

Jean-François Alcover, Oct 16 2013

Keywords

• nonn
• frac
• easy

Links

• Jean-François Alcover, Table of n, a(n) for n = 0..1000
• Peter Bala, A note on the sequence of numerators of a rational function
• Wikipedia, Quasi-polynomial

Crossrefs

Cf. A125650 (numerators).

Programs

Maple

seq( 4*(n + 1)*(n + 2)/igcd(4*(n + 1)*(n + 2), n*(n + 3)), n = 0..100); # Peter Bala, Feb 26 2019

Mathematica

Table[Denominator[n*(n+3)/(4*(n+1)*(n+2))], {n, 0, 100}]

PARI

vector(100, n, denominator((n-1)*(n+2)/(4*n*(n+1)))) \\ Colin Barker, Oct 09 2014

Formula

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]

From Peter Bala, Feb 26 2019: (Start)

a(n) = 4*(n + 1)*(n + 2)/gcd(4*(n + 1)*(n + 2), n*(n + 3)).

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)