A143601 Number of labeled odd-degree trees with 2n+1 nodes.

1, 1, 13, 541, 47545, 7231801, 1695106117, 567547087381, 257320926233329, 151856004814953841, 113144789723082206461, 103890621918675777804301, 115270544419577901796226473, 152049571406030636219959644841

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..211

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! satisfies the following formulas.

(1) A(x) = cosh(x*A(x)).

(2) A(x) = (1/x)*Series_Reversion( x/cosh(x) ).

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

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

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

(6) A(2*x) = [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)

(7) A(x/cosh(x)) = cosh(x).

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

(9) a(n) = Sum_{k=0..n} binomial(2*n+1,k) * (2*n+1 - 2*k)^(2*n) / ((2*n+1) * 2^(2*n)). [See formula by Christophe Vignat in A309204.] - Paul D. Hanna, Feb 19 2024

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

Radius of convergence r = 0.66274341934918158097474... = 1/sqrt(s^2-1) and A(r) = s (given above) satisfy r = 1/sinh(r*A(r)) and A(r) = cosh(r*A(r)). - Paul D. Hanna, Mar 04 2024

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
(PARI) {a(n) = sum(k=0, n, binomial(2*n+1, k) * (2*n+1-2*k)^(2*n) / ((2*n+1) * 2^(2*n)) )}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 19 2024

Crossrefs

Cf. A058014, A143600, A007106, A309204.

Cf. A370430, A370432.

Sequence in context: A309204 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