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%I #35 Mar 24 2023 16:59:54
%S 1,1,1,2,3,6,12,24,60,120,360,720,2520,5040,20160,40320,181440,362880,
%T 1814400,3628800,19958400,39916800,239500800,479001600,3113510400,
%U 6227020800,43589145600,87178291200,653837184000,1307674368000,10461394944000,20922789888000
%N G.f.: Sum_{n>=0} (1 + n*x)^n * x^n / (1 + x + n*x^2)^n.
%C 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
%F G.f.: 1/2 + (1+2*x) * Sum_{n>=0} (n+1)!*x^(2*n)/2.
%F a(2*n) = (n+1)!/2, a(2*n-1) = n!, for n>0 with a(0)=1.
%F From _Amiram Eldar_, Dec 11 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 3*e - 4.
%F Sum_{n>=0} (-1)^n/a(n) = e - 2. (End)
%e 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 +...
%e where
%e 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 +...
%t a[n_] := If[OddQ[n], ((n + 1)/2)!, (n/2 + 1)!/2]; a[0] = 1; Array[a, 32, 0] (* _Amiram Eldar_, Dec 11 2022 *)
%o (PARI) {a(n)=polcoeff( sum(m=0, n, (x+m*x^2)^m / (1 + x+m*x^2 +x*O(x^n))^m), n)}
%o for(n=0, 40, print1(a(n), ", "))
%o (PARI) {a(n)=if(n==0,1, if(n%2==0, ((n+2)/2)!/2, ((n+1)/2)! ))}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A000142, A001710, A187742, A187735, A208236, A204064, A361382.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jan 03 2013
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