A019583 a(n) = n*(n-1)^4/2.

0, 0, 1, 24, 162, 640, 1875, 4536, 9604, 18432, 32805, 55000, 87846, 134784, 199927, 288120, 405000, 557056, 751689, 997272, 1303210, 1680000, 2139291, 2693944, 3358092, 4147200, 5078125, 6169176, 7440174, 8912512, 10609215, 12555000, 14776336, 17301504, 20160657

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) = n(n-1)^4/2 is half the number of colorings of 5 points on a line with n colors. - R. H. Hardin, Feb 23 2002

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A101362.

Programs

Magma

[n*(n-1)^4/2: n in [0..30]]; // Vincenzo Librandi, Apr 20 2012

Mathematica

CoefficientList[Series[x^2*(1+18*x+33*x^2+8*x^3)/(1-x)^6,{x,0,40}],x] (* Vincenzo Librandi, Apr 20 2012 *)
a[n_] := n*(n - 1)^4/2; Array[a, 30, 0] (* Amiram Eldar, Feb 13 2023 *)

Formula

Sum_{j>=2} 1/a(j) = hypergeom([1, 1, 1, 1, 1], [ 2, 2, 2, 3], 1) = -2 + 2*zeta(2) - 2*zeta(3) + 2*zeta(4). - Stephen Crowley, Jun 28 2009

G.f.: x^2*(1 + 18*x + 33*x^2 + 8*x^3)/(1 - x)^6. - Colin Barker, Feb 23 2012

From Amiram Eldar, Feb 13 2023: (Start)

a(n) = A101362(n-1)/2.

Sum_{n>=2} (-1)^n/a(n) = 2 + Pi^2/6 + 7*Pi^4/360 - 4*log(2) - 3*zeta(3)/2. (End)