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A001472 Number of degree-n permutations of order dividing 4.
(Formerly M1292 N0495)
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 (list; graph; refs; listen; history; text; internal format)
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+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. [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. [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-Fran├žois 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 */

CROSSREFS

Cf. A000085, A001470, A053495.

Column k=4 of A008307.

Sequence in context: A255730 A010362 A262164 * 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

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Last modified October 19 16:07 EDT 2018. Contains 316366 sequences. (Running on oeis4.)