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URL: https://oeis.org/A001472

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A001472
Number of degree-n permutations of order dividing 4.
39
1, 1, 2, 4, 16, 56, 256, 1072, 6224, 33616, 218656, 1326656, 9893632, 70186624, 574017536, 4454046976, 40073925376, 347165733632, 3370414011904, 31426411211776, 328454079574016, 3331595921852416, 37125035407900672, 400800185285464064, 4744829049220673536
OFFSET
0,3
REFERENCES
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.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..570 (first 201 terms from T. D. Noe)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 25 (Dead link)
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.
FORMULA
E.g.f.: exp(x + x^2/2 + x^4/4).
D-finite with recurrence: a(0)=1, a(1)=1, a(2)=2, a(3)=4, a(n) = a(n-1) + (n-1)*a(n-2) + (n^3-6*n^2+11*n-6)*a(n-4) for n>3. - H. Palsdottir (hronn07(AT)ru.is), Sep 19 2008
a(n) = n!*Sum_{k=1..n} (1/k!)*Sum_{j=floor((4*k-n)/3)..k} binomial(k,j) * binomial(j,n-4*k+3*j) * (1/2)^(n-4*k+3*j)*(1/4)^(k-j), n>0. - Vladimir Kruchinin, Sep 07 2010
a(n) ~ n^(3*n/4)*exp(n^(1/4)-3*n/4+sqrt(n)/2-1/8)/2 * (1 - 1/(4*n^(1/4)) + 17/(96*sqrt(n)) + 47/(128*n^(3/4))). - Vaclav Kotesovec, Aug 09 2013
MAPLE
spec := [S, {S = Set(Union(Cycle(Z, card = 1), Cycle(Z, card = 2), Cycle(Z, card = 4)))}, labeled]; seq(combstruct[count](spec, size = n), n = 0 .. 23); # David Radcliffe, Aug 29 2025
MATHEMATICA
n = 23; CoefficientList[Series[Exp[x+x^2/2+x^4/4], {x, 0, n}], x] * Table[k!, {k, 0, n}] (* Jean-François Alcover, May 18 2011 *)
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); /* Vladimir Kruchinin, Sep 07 2010 */
(PARI) my(N=33, x='x+O('x^N)); egf=exp(x+x^2/2+x^4/4); Vec(serlaplace(egf)) /* Joerg Arndt, Sep 15 2012 */
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 +x^4/4) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 14 2019
(SageMath) m = 30; T = taylor(exp(x + x^2/2 + x^4/4), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 14 2019
CROSSREFS
Column k=4 of A008307.
Sequence in context: A262164 A322940 A306519 * A053498 A005388 A053503
KEYWORD
nonn,nice,easy
STATUS
approved