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A195865 G.f. satisfies: A(x) = exp( Sum_{n>=1} (A(x^n) + A(-x^n))/2 * x^n/n ). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..300

David Callan, Even outdegree rooted trees, (7-July-2014).

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).

EXAMPLE

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)

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 *)

PROG

(PARI) {a(n)=local(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)}

CROSSREFS

Sequence in context: A306318 A091188 A147678 * A222006 A127712 A305840

Adjacent sequences:  A195862 A195863 A195864 * A195866 A195867 A195868

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 26 2011

STATUS

approved

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Last modified August 15 16:02 EDT 2020. Contains 336505 sequences. (Running on oeis4.)