A371210 - OEIS

%I #20 Mar 25 2024 08:40:45

%S 1,1,4,2,18,6,96,24,600,120,4320,720,35280,5040,322560,40320,3265920,

%T 362880,36288000,3628800,439084800,39916800,5748019200,479001600,

%U 80951270400,6227020800,1220496076800,87178291200,19615115520000,1307674368000,334764638208000

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

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumVertexColoring.html">Minimum Vertex Coloring</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PathComplementGraph.html">Path Complement Graph</a>.

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

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

%F 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

%F 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

%p A371210 := proc(n)

%p     if type(n,'odd') then

%p         (n+1)/2*factorial((n+1)/2) ;

%p     else

%p         factorial(n/2) ;

%p     end if;

%p end proc:

%p seq(A371210(n),n=1..40) ; # _R. J. Mathar_, Mar 25 2024

%t Table[Piecewise[{{(n + 1)/2 ((n + 1)/2)!, Mod[n, 2] == 1}}, (n/2)!], {n, 31}]

%t 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]!

%o (Python)

%o from math import factorial

%o def A371210(n): return (m:=n+1>>1)*factorial(m) if n&1 else factorial(n>>1) # _Chai Wah Wu_, Mar 15 2024

%Y Cf. A001563, A000142.

%K nonn,easy

%O 1,3

%A _Eric W. Weisstein_, Mar 15 2024