A274276 E.g.f. A(x) satisfies: A( sqrt( A(x^2*exp(-2*x)) ) ) = x, where A(x) = Sum_{n>=1} a(n)*x^n/(n-1)!.

1, 1, 2, 10, 80, 776, 8992, 130768, 2252672, 43823872, 957193856, 23369928704, 629680631296, 18514472015872, 590350181439488, 20311856724176896, 749913022501879808, 29561045244530032640, 1239353203580700000256, 55077035791625925492736, 2586090541400666789543936, 127922890235433583945056256, 6649362432158408977810522112, 362360171399316029979428126720, 20658795751396952768159379619840

Offset

1,3

Links

Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 1..200 (first 150 terms from Paul D. Hanna)

Formula

a(n) = A274275(n)/n.

E.g.f. A(x) = Sum_{n>=1} a(n) * x^n / (n-1)! satisfies:

(1) A( sqrt( A(x^2*exp(2*x)) ) ) = -LambertW(-x*exp(x)).

(2) A(x) = Series_Reversion( sqrt( A(x^2*exp(-x)) ) ).

(3) A( A(x)^2 * exp(-2*A(x)) ) = x^2.

(4) A(-A(x)^2 * exp(-2*A(x)) ) = -LambertW(x^2*exp(-x^2)).

a(n)/n! ~ c * d^n / n^(5/2), where d = 2.52462188117..., c = 0.36965356... . - Vaclav Kotesovec, Jun 23 2016

Example

E.g.f.: A(x) = x + x^2 + 2*x^3/2! + 10*x^4/3! + 80*x^5/4! + 776*x^6/5! + 8992*x^7/6! + 130768*x^8/7! + 2252672*x^9/8! + 43823872*x^10/9! + 957193856*x^11/10! +...
such that A( sqrt( A(x^2*exp(-2*x)) ) ) = x.

Prog

(PARI) {a(n) = my(A=x); for(i=1, n, A = serreverse( sqrt( subst(A, x, x^2*exp(-2*x +x*O(x^n))) ) ) ); (n-1)!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A274275.

Sequence in context: A108486 A152168 A003578 * A152600 A371460 A220112

Adjacent sequences: A274273 A274274 A274275 * A274277 A274278 A274279

Keyword

nonn

Author

Paul D. Hanna, Jun 17 2016

Status

approved