A277168 Coefficients in the series reversion of x*exp(-x^2).

1, 6, 300, 41160, 11022480, 4870182240, 3211179491520, 2955402450000000, 3619848890071814400, 5693251850259515942400, 11182902317022859155532800, 26829777470359851910918195200, 77205601373291015625000000000000, 262462731609500617105685266652160000, 1040649406375513845021644881423595520000

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

G. C. Greubel, Table of n, a(n) for n = 1..195

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

Adjacent sequences: A277165 A277166 A277167 * A277169 A277170 A277171

Keyword

nonn

Author

Paul D. Hanna, Oct 01 2016

Status

approved