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A060435 Number of functions f: {1,2,...,n} -> {1,2,...,n} with even cycles only. 12
1, 0, 1, 6, 57, 680, 9945, 172032, 3438673, 78003648, 1980083025, 55616359040, 1712630427849, 57375166877184, 2077563829893097, 80859304977696000, 3366275257190794785, 149270897223530835968, 7024011523121427204897, 349574012216588890718208 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

E.g.f. equals the square-root of the e.g.f. of A134095. - Paul D. Hanna, Oct 11 2007

REFERENCES

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..386

FORMULA

E.g.f.: 1/sqrt(1-(LambertW(-x))^2). a(n)=(n-1)!*Sum_{k=0..floor((n-2)/2)} (k+1)/2^(2*k+1)*binomial(2*k+2, k+1)*n^(n-2-2*k)/(n-2-2*k)!.

A134095(n) = Sum_{k=0..n} C(n,k) * a(n-k) * a(k) with a(0)=1 and a(1)=0 where A134095(n) = Sum_{k=0..n} C(n,k) * (n-k)^k * k^(n-k). - Paul D. Hanna, Oct 11 2007

a(n) ~ n! * 2^(3/4)*Gamma(3/4)*exp(n)/(4*Pi*n^(3/4)) * (1- 5*Pi/ (24*Gamma(3/4)^2*sqrt(n))). - Vaclav Kotesovec, Sep 24 2013

EXAMPLE

E.g.f. A(x) = 1 + 0*x + 1*x^2/2! + 6*x^3/3! + 57*x^4/4! + 680*x^5/5! +...

The formula A(x) = 1/sqrt(1 - LambertW(-x)^2 ) is illustrated by:

A(x) = 1/sqrt(1 - (x+ x^2+ 3^2*x^3/3!+ 4^3*x^4/4!+ 5^4*x^5/5! +...)^2).

MATHEMATICA

t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Range[0, 20]! CoefficientList[Series[(1/(1 - t^2))^(1/2), {x, 0, 20}], x]  (* Geoffrey Critzer, Dec 07 2011 *)

PROG

(PARI) {a(n)=local(LambertW=sum(k=0, n, (-x)^(k+1)*(k+1)^k/(k+1)!) +x*O(x^n)); n!*polcoeff(1/sqrt(1-subst(LambertW, x, -x)^2), n)} - Paul D. Hanna, Oct 11 2007

CROSSREFS

Cf. A134095.

Column k=2 of A246609.

Sequence in context: A124556 A207412 A324447 * A153851 A141372 A306030

Adjacent sequences:  A060432 A060433 A060434 * A060436 A060437 A060438

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Apr 07 2001

EXTENSIONS

More terms from Alois P. Heinz, Aug 26 2014

STATUS

approved

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Last modified March 26 01:08 EDT 2019. Contains 321479 sequences. (Running on oeis4.)