A195865 G.f. satisfies A(x) = exp( Sum_{n>=1} (A(x^n) + A(-x^n))/2 * x^n/n ).

1, 1, 1, 2, 2, 4, 5, 10, 12, 25, 33, 68, 91, 190, 264, 555, 780, 1649, 2365, 5021, 7274, 15518, 22727, 48646, 71784, 154162, 229094, 493346, 737215, 1591518, 2390072, 5170896, 7798020, 16903848, 25587218, 55561618, 84377881, 183509750, 279499063, 608726985, 929556155, 2027094432

Offset: 0

Author

Paul D. Hanna, Oct 26 2011

Keywords

• nonn

Comments

For n>=1, a(n) is the number of rooted trees (see A000081) with n non-root nodes where non-root nodes cannot have odd out-degrees, see the note by David Callan and the example. - Joerg Arndt, Jun 28 2014

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 10*x^7 +...
Let B(x) = (A(x) + A(-x))/2 then
log(A(x)) = B(x) + B(x^2)*x^2/2 + B(x^3)*x^3/3 + B(x^4)*x^4/4 +...
The coefficients in (A(x) + A(-x))/2 begin:
[1,0,1,0,2,0,5,0,12,0,33,0,91,0,264,0,780,0,2365,0,7274,...]
from which the Euler transform generates the g.f. A(x):
A(x) = 1/((1-x)*(1-x^3)*(1-x^5)^2*(1-x^7)^5*(1-x^9)^12*(1-x^11)^33*(1-x^13)^91*...*(1-x^(2*n+1))^a(2*n)*...).
From _Joerg Arndt_, Jun 28 2014: (Start)
The a(6) = 5 rooted trees with 6 non-root nodes as described in the comment are:
:           level sequence       out-degrees (dots for zeros)
:     1:  [ 0 1 2 3 3 2 1 ]    [ 2 2 2 . . . . ]
:  O--o--o--o
:        .--o
:     .--o
:  .--o
:
:     2:  [ 0 1 2 2 2 2 1 ]    [ 2 4 . . . . . ]
:  O--o--o
:     .--o
:     .--o
:     .--o
:  .--o
:
:     3:  [ 0 1 2 2 1 2 2 ]    [ 2 2 . . 2 . . ]
:  O--o--o
:     .--o
:  .--o--o
:     .--o
:
:     4:  [ 0 1 2 2 1 1 1 ]    [ 4 2 . . . . . ]
:  O--o--o
:     .--o
:  .--o
:  .--o
:  .--o
:
:     5:  [ 0 1 1 1 1 1 1 ]    [ 6 . . . . . . ]
:  O--o
:  .--o
:  .--o
:  .--o
:  .--o
:  .--o
:
(End)
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..300 from Vincenzo Librandi)
• David Callan, Even outdegree rooted trees, (7-July-2014).

Crossrefs

Cf. A115593.

Programs

Mathematica

a[1] = 1;
a[n_] := a[n] = 1/(n - 1) Sum[(2 m + 1) a[2 m + 1] a[n - k (2 m + 1)], {m, 0, Floor[n/2] - 1}, {k, Floor[(n - 1)/(2 m + 1)]}];
Table[a[n], {n, 30}] (* Use offset 1 to simplify defining equation for G.f. Then apply xD_x, simplify, and equate coefficients to get above recurrence. - David Callan, Jul 07 2014 *)

PARI

{a(n)=my(A=1+x,B); for(i=1,n, B=(A+subst(A,x,-x))/2; A=exp(sum(m=1,n,subst(B,x,x^m+x*O(x^n))*x^m/m))); polcoeff(A,n)}

Formula

Euler transform of the coefficients in (A(x) + A(-x))/2.

G.f. satisfies: A(x) = Product_{n>=0} 1/(1 - x^(2*n+1))^a(2*n).

G.f. satisfies: A(x)*A(-x) = A(x^2).

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k and d is odd} d * a(d-1) ) * a(n-k). - Seiichi Manyama, May 31 2023