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A001472
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Number of degree-n permutations of order dividing 4.
(Formerly M1292 N0495)
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38
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1, 1, 2, 4, 16, 56, 256, 1072, 6224, 33616, 218656, 1326656, 9893632, 70186624, 574017536, 4454046976, 40073925376, 347165733632, 3370414011904, 31426411211776, 328454079574016, 3331595921852416, 37125035407900672, 400800185285464064
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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REFERENCES
| L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.
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 5.2.10.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 25
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565
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FORMULA
| E.g.f.: exp(x+1/2*x^2+1/4*x^4).
a(0)=1, a(1)=1, a(2)=2, a(3)=4, a(n)=a(n-1)+(n-1)*a(n-2)+(n^3-6n^2+11n-6)*a(n-4) for n>3. [From H. Palsdottir (hronn07(AT)ru.is), Sep 19 2008]
a(n)=n!*sum(sum(binomial(k,j)*binomial(j,n-4*k+3*j)*(1/2)^(n-4*k+3*j)*(1/4)^(k-j),j,floor((4*k-n)/3),k)/k!,k,1,n), n>0. [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Sep 07 2010]
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MATHEMATICA
| n = 23; CoefficientList[Series[Exp[x+x^2/2+x^4/4], {x, 0, n}], x] * Table[k!, {k, 0, n}]
(* From Jean-François Alcover, May 18 2011 *)
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PROG
| (Maxima) a(n):=n!*sum(sum(binomial(k, j)*binomial(j, n-4*k+3*j)*(1/2)^(n-4*k+3*j)*(1/4)^(k-j), j, floor((4*k-n)/3), k)/k!, k, 1, n); [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Sep 07 2010]
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CROSSREFS
| Cf. A000085, A001470, A053495.
Sequence in context: A104354 A153948 A010362 * A053498 A005388 A053503
Adjacent sequences: A001469 A001470 A001471 * A001473 A001474 A001475
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com) and J. H. Conway (conway(AT)math.princeton.edu)
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