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A143601 Number of labeled odd-degree trees with 2n+1 nodes. 5
1, 1, 13, 541, 47545, 7231801, 1695106117, 567547087381, 257320926233329, 151856004814953841, 113144789723082206461, 103890621918675777804301, 115270544419577901796226473, 152049571406030636219959644841 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..13.

FORMULA

E.g.f. satisfies: A(x) = cosh(x*A(x)).

E.g.f.: A(x) = (1/x)*Series_Reversion( x/cosh(x) ).

E.g.f.: sqrt(A(x)^2 - 1) = e.g.f. of A007106.

E.g.f.: exp(x*A(x)) = A(x) + sqrt(A(x)^2-1) = e.g.f. of A058014.

E.g.f.: A(x) = [F(x) + F(-x)]/2 where F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014.

E.g.f.: A(2x) = [G(x)/G(-x) + G(-x)/G(x)]/2 where G(x) = exp(x*G(x)/G(-x)) = e.g.f. of A143600.

From Paul D. Hanna, Aug 29 2008: (Start)

E.g.f. satisfies: A(x/cosh(x)) = cosh(x).

a(n) = (2n)!*[x^(2n)] cosh(x)^(2n+1)/(2n+1). (End)

a(n) ~ 2^(2*n) * n^(2*n-1) * (s^2-1)^(n+1/2) / exp(2*n), where s = 1.810170580698977274512829... is the root of the equation sqrt(s^2-1) * log(s + sqrt(s^2-1)) = s. - Vaclav Kotesovec, Jan 10 2014

EXAMPLE

E.g.f.: A(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! + ...

The e.g.f. of A007106 (a bisection of A058014) is given by:

sqrt(A(x)^2 - 1) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! + ...

The e.g.f. of A058014 is given by:

F(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! + ...

where A(x) = [F(x) + F(-x)]/2 and exp(x*A(x)) = F(x).

The e.g.f. of A143600 is given by:

G(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 249*x^4/4! + 2561*x^5/5! + ...

where A(2x) = [G(x)/G(-x) + G(-x)/G(x)]/2.

MATHEMATICA

Table[(2*n)!*CoefficientList[1/x*InverseSeries[Series[x/Cosh[x], {x, 0, 41}], x], x][[2*n+1]], {n, 0, 20}] (* Vaclav Kotesovec, Jan 10 2014 *)

PROG

(PARI) {a(n)=local(A=1+x*O(x^n)); for(i=0, n, A=cosh(x*A)); n!*polcoeff(A, n)}

(PARI) {a(n)=(2*n)!*polcoeff(cosh(x+x*O(x^(2*n)))^(2*n+1)/(2*n+1), 2*n)} \\ Paul D. Hanna, Aug 29 2008

CROSSREFS

Cf. A058014, A143600, A007106.

Sequence in context: A023332 A229263 A308865 * A282837 A203360 A265503

Adjacent sequences:  A143598 A143599 A143600 * A143602 A143603 A143604

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 26 2008, May 27 2009

EXTENSIONS

Edited by Paul D. Hanna, May 27 2009

STATUS

approved

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Last modified July 24 04:08 EDT 2019. Contains 325290 sequences. (Running on oeis4.)