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%I #28 Aug 18 2014 15:06:47
%S 1,1,4,14,66,384,2640,20880,186480,1854720,20321280,243129600,
%T 3153427200,44068147200,660064204800,10548573235200,179151388416000,
%U 3222109642752000,61178237632512000,1222853377794048000,25667116186263552000,564433265896980480000,12977099311614197760000
%N G.f.: Sum_{n>=0} (n+x)^n * x^n / (1 + n*x + x^2)^n.
%C For values of n between 3 and 11 (possibly continuing) the number of conjugacy classes of the symmetric group S_n when conjugating by a single transposition. - _Attila Egri-Nagy_, Aug 15 2014
%F a(n) = (n^2+n+2) * (n-1)!/2, for n>1 with a(0)=a(1)=1.
%F E.g.f.: 1/2 + 1/(2*(1-x)^2) - x - log(1-x).
%F E.g.f.: Sum_{n>=0} a(n+1)*x^n/n! = 1/(1-x)^3 + x/(1-x).
%e G.f.: A(x) = 1 + x + 4*x^2 + 14*x^3 + 66*x^4 + 384*x^5 + 2640*x^6 +...
%e where
%e A(x) = 1 + (1+x)*x/(1+x+x^2) + (2+x)^2*x^2/(1+2*x+x^2)^2 + (3+x)^3*x^3/(1+3*x+x^2)^3 + (4+x)^4*x^4/(1+4*x+x^2)^4 + (5+x)^5*x^5/(1+5*x+x^2)^5 +...
%t a[0] = 1; a[1] = 1; a[n_] := (n^2 + n + 2)*(n - 1)!/2; Table[a[n], {n, 0, 20}] (* _Wesley Ivan Hurt_, Aug 15 2014 *)
%o (PARI) {a(n)=polcoeff( sum(m=0, n, (m+x)^m*x^m/(1+m*x+x^2 +x*O(x^n))^m), n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n)=if(n>=0&n<=1,1,(n^2+n+2)*(n-1)!/2)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n)=n!*polcoeff(1/2 + 1/(2*(1-x)^2) - x - log(1-x +x*O(x^n)), n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) x='x+O('x^66); concat([1], Vec(serlaplace(1/(1-x)^3 + x/(1-x)))) \\ _Joerg Arndt_, Aug 15 2014
%o (GAP) List([3..11], n->Size(OrbitsDomain(Group((1,2)),SymmetricGroup(IsPermGroup, n), \^)); # _Attila Egri-Nagy_, Aug 15 2014
%Y Cf. A202365, A187741, A187735, A187746.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 03 2013
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