|
%I #41 Jan 29 2024 08:59:47
%S 1,0,2,12,120,1480,22320,396564,8118656,188185680,4871980800,
%T 139342178140,4363291266048,148470651659928,5455056815237120,
%U 215238256785814500,9077047768435752960,407449611073696325536,19396232794530856894464,976025303642559490903980
%N Expansion of e.g.f. A(x) = 1/(1 - LambertW(-x)^2).
%C 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.
%H Vincenzo Librandi, <a href="/A134095/b134095.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{k=0..n} C(n,k) * (n-k)^k * k^(n-k).
%F a(n) = n!*Sum_{k=0..n} (-1)^(n-k)*n^k/k!. - _Vladeta Jovovic_, Oct 17 2007
%F a(n) ~ n^n/2. - _Vaclav Kotesovec_, Nov 27 2012, simplified Nov 22 2021
%F a(n) = n! * [x^n] exp(n*x)/(1 + x). - _Ilya Gutkovskiy_, Sep 18 2018
%F a(n) = (-1)^n*exp(-n)*Integral_{x=-n..oo} x^n*exp(-x) dx. - _Thomas Scheuerle_, Jan 29 2024
%e E.g.f.: A(x) = 1 + 0*x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1480*x^5/5! + ...
%e The formula A(x) = 1/(1 - LambertW(-x)^2) is illustrated by:
%e A(x) = 1/(1 - (x + x^2 + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! + ...)^2).
%p seq(simplify(GAMMA(n+1,-n)*(-exp(-1))^n),n=0..20); # _Vladeta Jovovic_, Oct 17 2007
%t CoefficientList[Series[1/(1-LambertW[-x]^2), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Nov 27 2012 *)
%t a[x0_] := x D[1/x Exp[x], {x, n}] x^n Exp[-x] /. x->x0
%t Table[a[n], {n, 0, 20}] (* _Gerry Martens_, May 05 2016 *)
%o (PARI) {a(n)=sum(k=0,n,(n-k)^k*k^(n-k)*binomial(n,k))}
%o (PARI) /* Generated by e.g.f. 1/(1 - LambertW(-x)^2 ): */
%o {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)}
%Y Cf. A060435; indirectly related: A062817, A132608.
%Y Cf. A063170, A277458, A277490, A277510.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 11 2007
| 