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A000702 a(n) = number of conjugacy classes in the alternating group A_n.
(Formerly M2307 N0910)
9
1, 3, 4, 5, 7, 9, 14, 18, 24, 31, 43, 55, 72, 94, 123, 156, 200, 254, 324, 408, 513, 641, 804, 997, 1236, 1526, 1883, 2308, 2829, 3451, 4209, 5109, 6194, 7485, 9038, 10871, 13063, 15654, 18738, 22365, 26665, 31716, 37682, 44669, 52887, 62494, 73767 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

REFERENCES

Girse, Robert D.; The number of conjugacy classes of the alternating group. Nordisk Tidskr. Informationsbehandling (BIT) 20 (1980), no. 4, 515-517.

M. Osima, On the irreducible representations of the symmetric group, Canad. J. Math., 4 (1952), 381-384.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=2..1000

Eric Weisstein's World of Mathematics, Alternating Group.

FORMULA

a(n) = (p(n) + 3Q(n))/2 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-Erdos-Turan]

a(n) = 2p(n) + 3*Sum_{r>=1} (-1)^r*p(n-2r^2). [Girse]

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]

EXAMPLE

G.f. = x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 9*x^7 + 14*x^8 + 18*x^9 + ...

MATHEMATICA

p = PartitionsP; q[n_] := SeriesCoefficient[ Product[ 1+x^(2k+1), {k, 0, n}], {x, 0, n}]; a[n_] := (p[n] + 3*q[n])/2; Table[a[n], {n, 2, 48}] (* Jean-Fran├žois Alcover, Feb 22 2012, after first formula *)

a[ n_] := SeriesCoefficient[ ( 1 / QPochhammer[ x] + 3 / QPochhammer[ x, -x] ) / 2 - (2 + 2 x), {x, 0, n}]; (* Michael Somos, May 28 2014 *)

PROG

(MAGMA) [ NumberOfClasses(Alt(n)) : n in [2..10] ]; /* A useful example of MAGMA code, but it is better to use one of the formulae as below: */ A000702:= func< n | 2*NumberOfPartitions(n)+3*(&+[(-1)^r*NumberOfPartitions(n-2*r^2): r in [1..Isqrt(n div 2)]]) >; [A000702(n): n in [2..48]]; // Jason Kimberley, Feb 01 2011

(PARI) default(seriesprecision, 99);

Vec((1/eta(x)+3*eta(x^2)^2/(eta(x)*eta(x^4)))/2-(2+2*x)) /* Joerg Arndt, Feb 02 2011 */

CROSSREFS

Cf. A073584.

Sequence in context: A067530 A082922 A036971 * A067526 A101760 A165713

Adjacent sequences:  A000699 A000700 A000701 * A000703 A000704 A000705

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 24 12:28 EDT 2014. Contains 248516 sequences.