A187741 G.f.: Sum_{n>=0} (1 + n*x)^n * x^n / (1 + x + n*x^2)^n.

1, 1, 1, 2, 3, 6, 12, 24, 60, 120, 360, 720, 2520, 5040, 20160, 40320, 181440, 362880, 1814400, 3628800, 19958400, 39916800, 239500800, 479001600, 3113510400, 6227020800, 43589145600, 87178291200, 653837184000, 1307674368000, 10461394944000, 20922789888000

Offset: 0

Author

Paul D. Hanna, Jan 03 2013

Keywords

• nonn

Comments

This is an enumeration of the disjoint union (with repetition) of A001710(n), for n > 0, and A000142(n), for n > 0. The first lists the orders of the alternating groups; the second lists the orders of the symmetric groups. - Hal M. Switkay, Mar 13 2023

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 12*x^6 + 24*x^7 + 60*x^8 +...
where
A(x) = 1 + (1+x)*x/(1+x+x^2) + (1+2*x)^2*x^2/(1+x+2*x^2)^2 + (1+3*x)^3*x^3/(1+x+3*x^2)^3 + (1+4*x)^4*x^4/(1+x+4*x^2)^4 + (1+5*x)^5*x^5/(1+x+5*x^2)^5 +...
		

Crossrefs

Cf. A000142, A001710, A187742, A187735, A208236, A204064, A361382.

Programs

Mathematica

a[n_] := If[OddQ[n], ((n + 1)/2)!, (n/2 + 1)!/2]; a[0] = 1; Array[a, 32, 0] (* Amiram Eldar, Dec 11 2022 *)

PARI

{a(n)=polcoeff( sum(m=0, n, (x+m*x^2)^m / (1 + x+m*x^2 +x*O(x^n))^m), n)}
for(n=0, 40, print1(a(n), ", "))

PARI

{a(n)=if(n==0,1, if(n%2==0, ((n+2)/2)!/2, ((n+1)/2)! ))}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: 1/2 + (1+2*x) * Sum_{n>=0} (n+1)!*x^(2*n)/2.

a(2*n) = (n+1)!/2, a(2*n-1) = n!, for n>0 with a(0)=1.

From Amiram Eldar, Dec 11 2022: (Start)

Sum_{n>=0} 1/a(n) = 3*e - 4.

Sum_{n>=0} (-1)^n/a(n) = e - 2. (End)