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%I #34 Sep 08 2022 08:46:17
%S 1,6,300,41160,11022480,4870182240,3211179491520,2955402450000000,
%T 3619848890071814400,5693251850259515942400,
%U 11182902317022859155532800,26829777470359851910918195200,77205601373291015625000000000000,262462731609500617105685266652160000,1040649406375513845021644881423595520000
%N Coefficients in the series reversion of x*exp(-x^2).
%C Central terms of triangle A201685.
%C a(n) is the number of connected endofunctions on {1,2,...,2*n-1} that have exactly n nodes in the unique cycle of its digraph representation.
%H G. C. Greubel, <a href="/A277168/b277168.txt">Table of n, a(n) for n = 1..195</a>
%F E.g.f. A(x) = Sum_{n>=1} a(n) * x^(2*n-1) / (2*n-1)! satisfies:
%F (1) A( x*exp(-x^2) ) = x.
%F (2) A(x) = x*exp(A(x)^2).
%F (3) A(x) = Sum_{n>=1} (2*n-1)^(n-2) * x^(2*n-1) / (n-1)!.
%F (4) A(x)^2 = Sum_{n>=1} (2*n)^(n-1) * x^(2*n) / n!.
%F (5) A(x) = sqrt( LambertW(-2*x^2)/(-2) ).
%F a(n) = binomial(2*n-1, n-1) * (2*n-1)^(n-2) * n!.
%F a(n) ~ 2^(3*n-5/2) * n^(2*n-2) / exp(n+1/2). - _Vaclav Kotesovec_, Oct 07 2016
%e E.g.f.: A(x) = x + 6*x^3/3! + 300*x^5/5! + 41160*x^7/7! + 11022480*x^9/9! + 4870182240*x^11/11! + 3211179491520*x^13/13! + 2955402450000000*x^15/15! +...
%e such that A( x*exp(-x^2) ) = x.
%e E.g.f. may also be written
%e A(x) = x + 3^0*x^3/1! + 5^1*x^5/2! + 7^2*x^7/3! + 9^3*x^9/4! + 11^4*x^11/5! + ...
%e which satisfies: A(x) = x*exp(A(x)^2).
%e The square of the e.g.f. may be expressed as:
%e A(x)^2 = 2^0*x^2/1! + 4^1*x^4/2! + 6^2*x^6/3! + 8^3*x^8/4! + 10^4*x^10/5! + ...
%e which equals LambertW(-2*x^2)/(-2).
%p seq( binomial(2*n-1, n-1)*(2*n-1)^(n-2)*n!, n=1..15); # _G. C. Greubel_, Jan 08 2020
%t Table[Binomial[2*n-1, n-1]*(2*n-1)^(n-2)*n!, {n, 1, 15}] (* _Vaclav Kotesovec_, Oct 07 2016 *)
%o (PARI) {a(n) = (2*n-1)! * polcoeff( serreverse(x*exp(-x^2 +O(x^(2*n)))),2*n-1)}
%o for(n=1,20,print1(a(n),", "))
%o (PARI) {a(n) = binomial(2*n-1, n-1) * (2*n-1)^(n-2) * n! }
%o for(n=1,20,print1(a(n),", "))
%o (Magma) [Binomial(2*n-1, n-1)*(2*n-1)^(n-2)*Factorial(n): n in [1..15]]; // _G. C. Greubel_, Jan 08 2020
%o (Sage) [binomial(2*n-1, n-1)*(2*n-1)^(n-2)*factorial(n) for n in (1..15)] # _G. C. Greubel_, Jan 08 2020
%o (GAP) List([1..15], n-> Binomial(2*n-1, n-1)*(2*n-1)^(n-2)*Factorial(n)); # _G. C. Greubel_, Jan 08 2020
%Y Cf. A201685.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Oct 01 2016
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