|
| |
|
|
A060435
|
|
Number of functions f:{1,2,...,n}->{1,2,...,n} with even cycles only.
|
|
5
| |
|
|
1, 6, 57, 680, 9945, 172032, 3438673, 78003648, 1980083025, 55616359040, 1712630427849, 57375166877184, 2077563829893097, 80859304977696000, 3366275257190794785, 149270897223530835968, 7024011523121427204897
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 2,2
|
|
|
COMMENTS
| E.g.f. equals the square-root of the e.g.f. of A134095. - Paul D. Hanna (pauldhanna(AT)juno.com), Oct 11 2007
|
|
|
REFERENCES
| I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.
|
|
|
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 (pauldhanna(AT)juno.com), Oct 11 2007
|
|
|
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 (pauldhanna(AT)juno.com), Oct 11 2007
|
|
|
CROSSREFS
| Cf. A134095.
Sequence in context: A138414 A130565 A124556 * A153851 A141372 A152170
Adjacent sequences: A060432 A060433 A060434 * A060436 A060437 A060438
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 07 2001
|
| |
|
|
