%I #20 May 09 2023 11:41:06

%S 1,1,8,48,504,4680,66240,856800,14515200,242040960,4775500800,

%T 95520902400,2175146265600,50438868480000,1292330988748800,

%U 34092378448128000,971277752180736000,28566680100102144000,896191466580393984000,29029508406664077312000

%N Commuting permutations: number of ordered triples of permutations f, g, h in Symm(n) which all commute.

%D a(1)-a(7) computed by John McKay, Sep 06 2003.

%H T. D. Noe, <a href="/A072169/b072169.txt">Table of n, a(n) for n = 0..200</a>

%F Equals A061256(n)*n!.

%t nn = 20; b = Table[DivisorSigma[1, n], {n, nn}]; Range[0, nn]! CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* _T. D. Noe_, Jun 19 2012 *)

%o (Magma) for n in {1 .. 5} do G := SymmetricGroup(n); t1 := 0; for g in G do for h in G do for i in G do if g*h eq h*g and g*i eq i*g and h*i eq i*h then t1 := t1+1; end if; end for; end for; end for; n, t1; end for;

%Y Column k=3 of A362827.

%Y Cf. A061256, A053529.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Sep 06 2003. More terms from A061256 from _N. J. A. Sloane_, Jun 13 2012