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%I M2307 N0910 #59 Sep 08 2022 08:44:28
%S 1,1,3,4,5,7,9,14,18,24,31,43,55,72,94,123,156,200,254,324,408,513,
%T 641,804,997,1236,1526,1883,2308,2829,3451,4209,5109,6194,7485,9038,
%U 10871,13063,15654,18738,22365,26665,31716,37682,44669,52887,62494,73767
%N a(n) is the number of conjugacy classes in the alternating group A_n.
%D Girse, Robert D.; The number of conjugacy classes of the alternating group. Nordisk Tidskr. Informationsbehandling (BIT) 20 (1980), no. 4, 515-517.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000702/b000702.txt">Table of n, a(n) for n = 1..1000</a> [a(1)=1 prepended by _Georg Fischer_, Sep 29 2020]
%H R. D. Girse, <a href="/A000702/a000702.pdf">The number of conjugacy classes of the alternating group</a>, Preprint and correspondence [Annotated scanned copy]
%H M. Osima, <a href="http://dx.doi.org/10.4153/CJM-1952-034-x">On the irreducible representations of the symmetric group</a>, Canad. J. Math., 4 (1952), 381-384.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlternatingGroup.html">Alternating Group.</a>
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%F a(n) = (p(n) + 3Q(n))/2 for n>1 where p(n) denotes the number of unrestricted partitions of n (A000041) and Q(n) the number of partitions of n into distinct odd parts (A000700). [Denes-Erdős-Turan]
%F a(n) = 2p(n) + 3*Sum_{r>=1} (-1)^r*p(n-2r^2) for n>1. [Girse]
%F Sum_{r>=0} (-1)^r*a(n-(3r^2 +- r)/2) = 3(-1)^t if n = 2t^2 or 0 otherwise, where p(u) and a(u) are taken as 0 unless u is a nonnegative integer and t = 1,2,3,... [Girse]
%e G.f. = x + x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 9*x^7 + 14*x^8 + 18*x^9 + ...
%t p = PartitionsP; q[n_] := SeriesCoefficient[ Product[ 1+x^(2k+1), {k, 0, n}], {x, 0, n}]; a[1]=1; a[n_] := (p[n] + 3*q[n])/2; Table[a[n], {n, 48}] (* _Jean-François Alcover_, Feb 22 2012, after first formula *)
%t a[ n_] := SeriesCoefficient[ ( 1 / QPochhammer[ x] + 3 / QPochhammer[ x, -x] ) / 2 - (2 + x), {x, 0, n}]; Table[a[n], {n, 1, 48}] (* _Michael Somos_, May 28 2014 *)
%o (Magma) [ NumberOfClasses(Alt(n)) : n in [1..10] ]; /* A useful example of MAGMA code, but it is better to use one of the formulas as below: */ A000702:= func< n | 2*NumberOfPartitions(n)+3*(&+[(-1)^r*NumberOfPartitions(n-2*r^2): r in [1..Isqrt(n div 2)]]) >; [1] cat [A000702(n): n in [2..48]]; // _Jason Kimberley_, Feb 01 2011
%o (PARI) default(seriesprecision,99);
%o Vec((1/eta(x)+3*eta(x^2)^2/(eta(x)*eta(x^4)))/2-(2+x)) /* _Joerg Arndt_, Feb 02 2011 */
%Y Cf. A073584.
%K nonn,nice,easy
%O 1,3
%A _N. J. A. Sloane_
%E a(1)=1 added by _N. J. A. Sloane_, Sep 14 2020
%E Follow-up corrections by _Andrey Zabolotskiy_, Sep 18 2020
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