A065513
Number of endofunctions of [n] with a cycle a->b->c->a and for all x in [n], some iterate f^k(x)=a.
Original entry on oeis.org
2, 24, 300, 4320, 72030, 1376256, 29760696, 720000000, 19292299290, 567575838720, 18197320924068, 631732166467584, 23613833496093750, 945755921747804160, 40410678374256222960, 1835086247681868693504, 88263072551692077310386, 4482662400000000000000000
Offset: 3
a(4)=24: 1->2->3->1<-4; 2->3->1->2<-4; 3->1->2->3<-4 1->3->2->1<-4; 3->2->1->3<-4; 2->1->3->2<-4 (repeat with 1,2, then 3 excluded from cycle)
Cf.
A000169 (unique cycle is length 1),
A053506 (unique cycle has length 2).
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[(n-1)*(n-2)*n^(n-3): n in [3..50]]; // G. C. Greubel, Nov 14 2017
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T := x->-LambertW(-x); a := []; f := series((T(x))^3/3,x,24); for m from 1 to 24 do a := [op(a),op(2*m-1,f)*(m+2)! ] od; print(a);
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nn = 18; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
Range[0, nn]! CoefficientList[Series[2 t^3/3!, {x, 0, nn}], x] (* Geoffrey Critzer, Aug 14 2013 *)
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for(n=3,50, print1((n-1)*(n-2)*n^(n-3), ", ")) \\ G. C. Greubel, Nov 14 2017
A277168
Coefficients in the series reversion of x*exp(-x^2).
Original entry on oeis.org
1, 6, 300, 41160, 11022480, 4870182240, 3211179491520, 2955402450000000, 3619848890071814400, 5693251850259515942400, 11182902317022859155532800, 26829777470359851910918195200, 77205601373291015625000000000000, 262462731609500617105685266652160000, 1040649406375513845021644881423595520000
Offset: 1
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! +...
such that A( x*exp(-x^2) ) = x.
E.g.f. may also be written
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! + ...
which satisfies: A(x) = x*exp(A(x)^2).
The square of the e.g.f. may be expressed as:
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! + ...
which equals LambertW(-2*x^2)/(-2).
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List([1..15], n-> Binomial(2*n-1, n-1)*(2*n-1)^(n-2)*Factorial(n)); # G. C. Greubel, Jan 08 2020
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[Binomial(2*n-1, n-1)*(2*n-1)^(n-2)*Factorial(n): n in [1..15]]; // G. C. Greubel, Jan 08 2020
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seq( binomial(2*n-1, n-1)*(2*n-1)^(n-2)*n!, n=1..15); # G. C. Greubel, Jan 08 2020
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Table[Binomial[2*n-1, n-1]*(2*n-1)^(n-2)*n!, {n, 1, 15}] (* Vaclav Kotesovec, Oct 07 2016 *)
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{a(n) = (2*n-1)! * polcoeff( serreverse(x*exp(-x^2 +O(x^(2*n)))),2*n-1)}
for(n=1,20,print1(a(n),", "))
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{a(n) = binomial(2*n-1, n-1) * (2*n-1)^(n-2) * n! }
for(n=1,20,print1(a(n),", "))
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[binomial(2*n-1, n-1)*(2*n-1)^(n-2)*factorial(n) for n in (1..15)] # G. C. Greubel, Jan 08 2020
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