A371210 Number of minimum vertex colorings in the complement of the path graph on n nodes.

1, 1, 4, 2, 18, 6, 96, 24, 600, 120, 4320, 720, 35280, 5040, 322560, 40320, 3265920, 362880, 36288000, 3628800, 439084800, 39916800, 5748019200, 479001600, 80951270400, 6227020800, 1220496076800, 87178291200, 19615115520000, 1307674368000, 334764638208000

Offset

1,3

Links

Table of n, a(n) for n=1..31.

Eric Weisstein's World of Mathematics, Minimum Vertex Coloring.

Eric Weisstein's World of Mathematics, Path Complement Graph.

Formula

a(n) = (n+1)/2*((n+1)/2)! for n odd.

a(n) = (n/2)! for n even.

E.g.f.: (2*(6*x + x^3) + exp(x^2/4)*sqrt(Pi)*(4 + x*(8 + 8*x + x^3))*erf(x/2))/16. - Stefano Spezia, Mar 15 2024

D-finite with recurrence +2*(-29*n+56)*a(n) +2*(9*n-79)*a(n-1) +(29*n^2+31*n-52)*a(n-2) -(n-1)*(9*n-52)*a(n-3)=0. - R. J. Mathar, Mar 25 2024

Maple

A371210 := proc(n)
    if type(n, 'odd') then
        (n+1)/2*factorial((n+1)/2) ;
    else
        factorial(n/2) ;
    end if;
end proc:
seq(A371210(n), n=1..40) ; # R. J. Mathar, Mar 25 2024

Mathematica

Table[Piecewise[{{(n + 1)/2 ((n + 1)/2)!, Mod[n, 2] == 1}}, (n/2)!], {n, 31}]
CoefficientList[Series[(2 (6 x + x^3) + Exp[x^2/4] Sqrt[Pi] (4 + x (8 + 8 x + x^3)) Erf[x/2])/16, {x, 0, 20}], x] Range[0, 20]!

Prog

(Python)
from math import factorial
def A371210(n): return (m:=n+1>>1)*factorial(m) if n&1 else factorial(n>>1) # Chai Wah Wu, Mar 15 2024

Crossrefs

Cf. A001563, A000142.

Sequence in context: A328695 A285595 A255566 * A302461 A303243 A074676

Adjacent sequences: A371207 A371208 A371209 * A371211 A371212 A371213

Keyword

nonn,easy

Author

Eric W. Weisstein, Mar 15 2024

Status

approved