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A181954 Weighted sum of all cyclic subgroups of prime order of the symmetric group S_n.

%I A181954 #19 Jul 07 2018 01:39:02
%S A181954 0,2,9,30,110,450,2457,11774,47910,264630,5565065,56021922,866143434,
%T A181954 9894742130,78233264865,470058202350,24530748587102,409761902222094,
%U A181954 10595012400106545,160826238368038490,1585844131838898330,16787211702213659322,1362379222505265018329
%N A181954 Weighted sum of all cyclic subgroups of prime order of the symmetric group S_n.
%C A181954 Sum of the orders of all subgroups of prime order in S_n.
%H A181954 Stephen A. Silver, <a href="/A181954/b181954.txt">Table of n, a(n) for n = 1..451</a>
%F A181954 a(n) = A186202(n) + A214003(n).
%e A181954 The symmetric group S_3 has one subgroup of order 3 and three subgroups of order 2, and no other subgroups of prime order. So a(3) = 3 + 2 + 2 + 2 = 9.
%t A181954 a[n_] := Sum[If[PrimeQ[p], Sum[n!/(k!*(n-k*p)!*p^k), {k, 1, n/p}]*p/(p-1), 0], {p, 2, n}];
%t A181954 Array[a, 24] (* _Jean-François Alcover_, Jul 06 2018, after _Andrew Howroyd_ *)
%o A181954 (PARI) a(n)={sum(p=2, n, if(isprime(p), sum(k=1, n\p, n!/(k!*(n-k*p)!*p^k))*p/(p-1)))} \\ _Andrew Howroyd_, Jul 03 2018
%Y A181954 Cf. A181955 (the alternating case).
%K A181954 nonn
%O A181954 1,2
%A A181954 _Olivier Gérard_, Apr 03 2012
%E A181954 More terms from _Stephen A. Silver_, Feb 16 2013