A199202 E.g.f. satisfies: A(x) = exp( x*(A(x) + 1/A(-x))/2 ).

1, 1, 3, 10, 53, 376, 3607, 38032, 498409, 7122304, 121691051, 2182921984, 45592175389, 987527547904, 24479592884671, 620921169012736, 17795726532904913, 517636848366223360, 16851227968120051027, 552890360903850459136, 20150074601540899828741

Offset: 0

Author

Paul D. Hanna, Nov 03 2011

Keywords

• nonn

Comments

Compare to the e.g.f. G(x) of A058014, which satisfies both: G(x) = exp(x*(G(x) + 1/G(x))/2) and G(x) = exp(x*(G(x) + G(-x))/2); A058014 counts labeled trees such that the degrees of all nodes, excluding the first, are odd.

Examples

			E.g.f.:  A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 376*x^5/5! +.. .
Let B(x) = log(A(x))/x = (A(x) + 1/A(-x))/2 then B(x) begins:
B(x) = 1 + x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 216*x^5/5! + 1561*x^6/6! + 19328*x^7/7! +...+ A198198(n)*x^n/n! +...
such that B(x) = (exp(x*B(x)) + exp(x*B(-x)))/2.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..320

Crossrefs

Cf. A198198, A058014.

Programs

PARI

{a(n)=local(A=1+x*O(x^n)); for(n=0, n, A=exp(x*(A+1/subst(A, x, -x))/2+x*O(x^n))); n!*polcoeff(A, n)}

Formula

E.g.f.: A(x) = exp(x*B(x)) where B(x) = (exp(x*B(x)) + exp(x*B(-x)))/2 is the e.g.f. of A198198.

E.g.f. satisfies: log(x) = x*log(y)/(x*y^2 - 2*y*log(y)) + log(2*log(y) - x*y), where y = A(x). - Vaclav Kotesovec, Feb 28 2014

a(n) ~ c * n! * d^n / n^(3/2), where d = 1.9126860724609002014... (see A198198), and c = 1.84843299011729... if n is even, and c = 1.808309580980992... if n is odd. - Vaclav Kotesovec, Feb 28 2014