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A002135
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Number of terms in a symmetrical determinant: a(n) = n*a(n-1) - (n-1)*(n-2)*a(n-3)/2.
(Formerly M1513 N0594)
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7
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1, 1, 2, 5, 17, 73, 388, 2461, 18155, 152531, 1436714, 14986879, 171453343, 2134070335, 28708008128, 415017867707, 6416208498137, 105630583492969, 1844908072865290, 34071573484225549, 663368639907213281, 13580208904207073801
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n) is the number of collections of necklaces created by using exactly n different colored beads (to make the entire collection). [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Apr 19 2009]
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REFERENCES
| A. C. Aitken, On the number of distinct terms in the expansion of symmetric and skew determinants, Edinburgh Math. Notes, No. 34 (1944), 1-5.
A. Cayley, On the number of distinct terms in a symmetrical or partially symmetrical determinant, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 190.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260, #12, a_n.
P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41.
Problem E2297, Amer. Math. Monthly, 79 (1972), 519-520.
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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.9 and Problem 5.22.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
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FORMULA
| E.g.f.: (1-x)^(-1/2)*exp(x/2+x^2/4). a(n+1) = (n+1)*a(n) - binomial(n, 2)*a(n-2) - Comtet.
Asymptotics: a(n):=sqrt(2) exp(3/4 -n ) n^n (1+O(1/n)) [From Pietro Majer (majer(AT)dm.unipi.it), Oct 27 2009]
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MATHEMATICA
| a[x_]:=Log[1/(1-x)^(1/2)]+x/2+x^2/4; Range[0, 20]! CoefficientList[Series[Exp[a[x]], {x, 0, 20}], x]
RecurrenceTable[{a[0]==a[1]==1, a[2]==2, a[n]==n*a[n-1]-(n-1)(n-2)* a[n-3]/2}, a, {n, 30}] (* From Harvey P. Dale, Dec 16 2011 *)
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CROSSREFS
| Cf. A059422, A059423, A059424.
Sequence in context: A007779 A084161 A102038 * A007868 A136726 A112831
Adjacent sequences: A002132 A002133 A002134 * A002136 A002137 A002138
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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