A001472 Number of degree-n permutations of order dividing 4. (Formerly M1292 N0495)

1, 1, 2, 4, 16, 56, 256, 1072, 6224, 33616, 218656, 1326656, 9893632, 70186624, 574017536, 4454046976, 40073925376, 347165733632, 3370414011904, 31426411211776, 328454079574016, 3331595921852416, 37125035407900672, 400800185285464064

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

T. D. Noe, Table of n, a(n) for n=0..200

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 25

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

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) 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
(Sage) 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

Cf. A000085, A001470, A053495.

Column k=4 of A008307.

Sequence in context: A262164 A322940 A306519 * A053498 A005388 A053503

Adjacent sequences: A001469 A001470 A001471 * A001473 A001474 A001475

Keyword

nonn,nice,easy

Author

N. J. A. Sloane and J. H. Conway

Status

approved