A134095 Expansion of e.g.f. A(x) = 1/(1 - LambertW(-x)^2).

1, 0, 2, 12, 120, 1480, 22320, 396564, 8118656, 188185680, 4871980800, 139342178140, 4363291266048, 148470651659928, 5455056815237120, 215238256785814500, 9077047768435752960, 407449611073696325536, 19396232794530856894464, 976025303642559490903980

Offset: 0

Author

Paul D. Hanna, Oct 11 2007

Keywords

• nonn

Comments

E.g.f. equals the square of the e.g.f. of A060435, where A060435(n) = number of functions f: {1,2,...,n} -> {1,2,...,n} with even cycles only.

Examples

			E.g.f.: A(x) = 1 + 0*x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1480*x^5/5! + ...
The formula A(x) = 1/(1 - LambertW(-x)^2) is illustrated by:
A(x) = 1/(1 - (x + x^2 + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! + ...)^2).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A060435; indirectly related: A062817, A132608.

Cf. A063170, A277458, A277490, A277510.

Programs

Maple

seq(simplify(GAMMA(n+1,-n)*(-exp(-1))^n),n=0..20); # Vladeta Jovovic, Oct 17 2007

Mathematica

CoefficientList[Series[1/(1-LambertW[-x]^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
a[x0_] := x D[1/x Exp[x], {x, n}] x^n Exp[-x] /. x->x0
Table[a[n], {n, 0, 20}] (* Gerry Martens, May 05 2016 *)

PARI

{a(n)=sum(k=0,n,(n-k)^k*k^(n-k)*binomial(n,k))}

PARI

/* Generated by e.g.f. 1/(1 - LambertW(-x)^2 ): */
{a(n)=my(LambertW=-x*sum(k=0,n,(-x)^k*(k+1)^(k-1)/k!) +x*O(x^n)); n!*polcoeff(1/(1-subst(LambertW,x,-x)^2),n)}

Formula

a(n) = Sum_{k=0..n} C(n,k) * (n-k)^k * k^(n-k).

a(n) = n!*Sum_{k=0..n} (-1)^(n-k)*n^k/k!. - Vladeta Jovovic, Oct 17 2007

a(n) ~ n^n/2. - Vaclav Kotesovec, Nov 27 2012, simplified Nov 22 2021

a(n) = n! * [x^n] exp(n*x)/(1 + x). - Ilya Gutkovskiy, Sep 18 2018

a(n) = (-1)^n*exp(-n)*Integral_{x=-n..oo} x^n*exp(-x) dx. - Thomas Scheuerle, Jan 29 2024