A187742 G.f.: Sum_{n>=0} (n+x)^n * x^n / (1 + n*x + x^2)^n.

1, 1, 4, 14, 66, 384, 2640, 20880, 186480, 1854720, 20321280, 243129600, 3153427200, 44068147200, 660064204800, 10548573235200, 179151388416000, 3222109642752000, 61178237632512000, 1222853377794048000, 25667116186263552000, 564433265896980480000, 12977099311614197760000

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..22.

Formula

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

E.g.f.: 1/2 + 1/(2*(1-x)^2) - x - log(1-x).

E.g.f.: Sum_{n>=0} a(n+1)*x^n/n! = 1/(1-x)^3 + x/(1-x).

Example

G.f.: A(x) = 1 + x + 4*x^2 + 14*x^3 + 66*x^4 + 384*x^5 + 2640*x^6 +...
where
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 +...

Mathematica

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 *)

Prog

(PARI) {a(n)=polcoeff( sum(m=0, n, (m+x)^m*x^m/(1+m*x+x^2 +x*O(x^n))^m), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=if(n>=0&n<=1, 1, (n^2+n+2)*(n-1)!/2)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=n!*polcoeff(1/2 + 1/(2*(1-x)^2) - x - log(1-x +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) x='x+O('x^66); concat([1], Vec(serlaplace(1/(1-x)^3 + x/(1-x)))) \\ Joerg Arndt, Aug 15 2014
(GAP) List([3..11], n->Size(OrbitsDomain(Group((1, 2)), SymmetricGroup(IsPermGroup, n), \^)); # Attila Egri-Nagy, Aug 15 2014

Crossrefs

Cf. A202365, A187741, A187735, A187746.

Sequence in context: A241465 A320488 A126787 * A347432 A129219 A292719

Adjacent sequences: A187739 A187740 A187741 * A187743 A187744 A187745

Keyword

nonn

Author

Paul D. Hanna, Jan 03 2013

Status

approved