login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A277168 Coefficients in the series reversion of x*exp(-x^2). 2
1, 6, 300, 41160, 11022480, 4870182240, 3211179491520, 2955402450000000, 3619848890071814400, 5693251850259515942400, 11182902317022859155532800, 26829777470359851910918195200, 77205601373291015625000000000000, 262462731609500617105685266652160000, 1040649406375513845021644881423595520000 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Central terms of triangle A201685.
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.
LINKS
FORMULA
E.g.f. A(x) = Sum_{n>=1} a(n) * x^(2*n-1) / (2*n-1)! satisfies:
(1) A( x*exp(-x^2) ) = x.
(2) A(x) = x*exp(A(x)^2).
(3) A(x) = Sum_{n>=1} (2*n-1)^(n-2) * x^(2*n-1) / (n-1)!.
(4) A(x)^2 = Sum_{n>=1} (2*n)^(n-1) * x^(2*n) / n!.
(5) A(x) = sqrt( LambertW(-2*x^2)/(-2) ).
a(n) = binomial(2*n-1, n-1) * (2*n-1)^(n-2) * n!.
a(n) ~ 2^(3*n-5/2) * n^(2*n-2) / exp(n+1/2). - Vaclav Kotesovec, Oct 07 2016
EXAMPLE
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).
MAPLE
seq( binomial(2*n-1, n-1)*(2*n-1)^(n-2)*n!, n=1..15); # G. C. Greubel, Jan 08 2020
MATHEMATICA
Table[Binomial[2*n-1, n-1]*(2*n-1)^(n-2)*n!, {n, 1, 15}] (* Vaclav Kotesovec, Oct 07 2016 *)
PROG
(PARI) {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), ", "))
(PARI) {a(n) = binomial(2*n-1, n-1) * (2*n-1)^(n-2) * n! }
for(n=1, 20, print1(a(n), ", "))
(Magma) [Binomial(2*n-1, n-1)*(2*n-1)^(n-2)*Factorial(n): n in [1..15]]; // G. C. Greubel, Jan 08 2020
(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
(GAP) List([1..15], n-> Binomial(2*n-1, n-1)*(2*n-1)^(n-2)*Factorial(n)); # G. C. Greubel, Jan 08 2020
CROSSREFS
Cf. A201685.
Sequence in context: A264706 A066718 A065947 * A081321 A159494 A221850
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 01 2016
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 25 03:25 EDT 2024. Contains 374586 sequences. (Running on oeis4.)