login
Number of words of length 4 in the n(n-1)/2 transpositions of S[ n ] equivalent to the identity.
1

%I #14 Sep 08 2022 08:44:50

%S 0,1,27,120,340,765,1491,2632,4320,6705,9955,14256,19812,26845,35595,

%T 46320,59296,74817,93195,114760,139860,168861,202147,240120,283200,

%U 331825,386451,447552,515620,591165,674715,766816,868032,978945,1100155,1232280,1375956

%N Number of words of length 4 in the n(n-1)/2 transpositions of S[ n ] equivalent to the identity.

%H Vincenzo Librandi, <a href="/A029699/b029699.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = n*(n-1)*(3*n^2+n-12)/4.

%F a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>5. - _Colin Barker_, May 28 2015

%F G.f.: x^2*(5*x^2-22*x-1) / (x-1)^5. - _Colin Barker_, May 28 2015

%t LinearRecurrence[{5,-10,10,-5,1},{0,1,27,120,340},50] (* _Harvey P. Dale_, Jan 10 2022 *)

%o (Magma) [n*(n-1)*(3*n^2+n-12)/4: n in [1..45]]; // _Vincenzo Librandi_, Jun 30 2011

%o (PARI) concat(0, Vec(x^2*(5*x^2-22*x-1) / (x-1)^5 + O(x^100))) \\ _Colin Barker_, May 28 2015

%K nonn,easy

%O 1,3

%A Paolo Dominici (pl.dm(AT)libero.it)