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A087132
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a(n) is the sum of the squares of the sizes of the conjugacy classes in the symmetric group S_n.
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5
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1, 1, 2, 14, 146, 2602, 71412, 2675724, 134269158, 8747088662, 717107850956, 72007758701716, 8736187050160132, 1258160557017484564, 212232765513231245096, 41518913481377118146520, 9309797624034705006898470, 2374942651509463493006400390, 683620331016710787068868581580
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OFFSET
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0,3
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COMMENTS
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This is a natural quantity to consider when viewing the symmetric group (Sym_n) as a set. a(n) is the sum over all elements of Sym_n of the size of their conjugacy class. Each conjugacy class is thus counted as many times as its size, giving a sum of squares. - Olivier Gérard, Feb 12 2012
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LINKS
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FORMULA
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a(n) = (n!)^2 * (c/n^2 + O((log n)/n^3)), where c = prod_{k>=1}sum_{n>=0}1/(k*n!)^2 ~ 4.263403514152669778298935... (see A246879). [Corrected by Vaclav Kotesovec, Sep 21 2014]
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MAPLE
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b:= proc(n, i) option remember; uses combinat; `if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, i-1)*((i-1)!^j/j!*
multinomial(n, n-i*j, i$j, 0))^2, j=0..n/i)))
end:
a:= n-> b(n$2):
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MATHEMATICA
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multinomial[n_, k_List] := n! / Times @@ (k!);
b[n_, i_] := b[n, i] = If[n == 0, 1,
If[i < 1, 0, Sum[b[n-i*j, i-1]*((i-1)!^j/j!*
multinomial[n, {n-i*j, Sequence@@Table[i, {j}], 0}])^2, {j, 0, n/i}]]];
a[n_] := b[n, n];
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PROG
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(Magma) [ &+[ c[2]^2 : c in ClassesData(Sym(n))] : n in [1..10]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 18 2003
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EXTENSIONS
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STATUS
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approved
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