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%I #23 Dec 22 2018 12:19:02
%S 0,0,4,22,166,1266,11166,106128,1122192,12809520,159451920,2128973760,
%T 30594214080,468275713920,7641089769600,131971588761600,
%U 2412294180710400,46422407927347200,940023724189132800,19949344876532736000,443393309963068416000,10288553164881868800000
%N Weight of the alternating group (A_n) in transpositions.
%C Sequences A067369, A067370 and A067318 are related: A067318 = A067369 + A067370. A067318 counts transpositions in the symmetric group, denoted S_n. One can think of the transpositions in S_n as being split between the alternating group A_n and its complement, which we call the periphery and denote P_N. For n >= 3, A067369 v(P_N) and A067370 v(A_n) always differ by (n-2)!. When n is odd, v(A_n) is larger; when n is even, v(P_N) is larger. This gives new meaning to the name alternating group. The average weight of permutation in A_n converges with the average weight for a permutation in P_N at infinity.
%H Charlie Neder and Muniru A Asiru, <a href="/A067369/b067369.txt">Table of n, a(n) for n = 1..445</a>
%F a(n) = a(n-1) + [(n-1)!/2]*[vbar(P_N-1)+1]*[n-1)] where vbar(P_N) is the average weight of a permutation in P_N, the periphery of A_n. vbar(P_N-1) is p(n-1)/(n-1)!2 where p(n) is from sequence A067370.
%F From _Vladeta Jovovic_, Feb 02 2003: (Start)
%F a(n) = (1/2)*((-1)^(n+1)*(n-2)! + n*n! - abs(Stirling1(n+1, 2))), n > 1.
%F E.g.f.: (1/2)*(-(1+x)*log(1+x) + x + x/(1-x)^2 + log(1-x)/(1-x) + 2). (End)
%p seq(coeff(series(factorial(n)*(1/2)*(-(1+x)*log(1+x)+x+x/(1-x)^2+log(1-x)/(1-x)+2),x,n+1), x, n), n = 1 .. 25); # _Muniru A Asiru_, Dec 15 2018
%t a[n_] := 1/2*((-1)^(n+1)*(n-2)!+n*n!-Abs[StirlingS1[n+1, 2]]); a[1]=0; Table[a[n], {n, 1, 20}] (* _Jean-François Alcover_, Jan 12 2015, after _Vladeta Jovovic_ *)
%o (PARI) a(n)={if(n < 2, 0, 1/2*((-1)^(n+1)*(n-2)!+n*n!-abs(stirling(n+1, 2, 1))))} \\ _Andrew Howroyd_, Dec 14 2018
%o (GAP) Concatenation([0],List([2..25],n->(1/2)*((-1)^(n+1)*Factorial(n-2)+n*Factorial(n)-AbsInt(Stirling1(n+1,2))))); # _Muniru A Asiru_, Dec 15 2018
%Y Cf. A067370, A067318.
%K easy,nice,nonn
%O 1,3
%A Nick Hann (nickhann(AT)aol.com), Jan 20 2002
%E More terms from _Vladeta Jovovic_, Feb 02 2003
%E a(20)-a(22) from _Charlie Neder_, Dec 14 2018
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