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%I #40 May 31 2023 14:33:23
%S 1,1,1,2,2,4,5,10,12,25,33,68,91,190,264,555,780,1649,2365,5021,7274,
%T 15518,22727,48646,71784,154162,229094,493346,737215,1591518,2390072,
%U 5170896,7798020,16903848,25587218,55561618,84377881,183509750,279499063,608726985,929556155,2027094432
%N G.f. satisfies A(x) = exp( Sum_{n>=1} (A(x^n) + A(-x^n))/2 * x^n/n ).
%C 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
%H Seiichi Manyama, <a href="/A195865/b195865.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..300 from Vincenzo Librandi)
%H David Callan, <a href="/A195865/a195865.pdf">Even outdegree rooted trees</a>, (7-July-2014).
%F Euler transform of the coefficients in (A(x) + A(-x))/2.
%F G.f. satisfies: A(x) = Product_{n>=0} 1/(1 - x^(2*n+1))^a(2*n).
%F G.f. satisfies: A(x)*A(-x) = A(x^2).
%F 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
%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 10*x^7 +...
%e Let B(x) = (A(x) + A(-x))/2 then
%e log(A(x)) = B(x) + B(x^2)*x^2/2 + B(x^3)*x^3/3 + B(x^4)*x^4/4 +...
%e The coefficients in (A(x) + A(-x))/2 begin:
%e [1,0,1,0,2,0,5,0,12,0,33,0,91,0,264,0,780,0,2365,0,7274,...]
%e from which the Euler transform generates the g.f. A(x):
%e 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)*...).
%e From _Joerg Arndt_, Jun 28 2014: (Start)
%e The a(6) = 5 rooted trees with 6 non-root nodes as described in the comment are:
%e : level sequence out-degrees (dots for zeros)
%e : 1: [ 0 1 2 3 3 2 1 ] [ 2 2 2 . . . . ]
%e : O--o--o--o
%e : .--o
%e : .--o
%e : .--o
%e :
%e : 2: [ 0 1 2 2 2 2 1 ] [ 2 4 . . . . . ]
%e : O--o--o
%e : .--o
%e : .--o
%e : .--o
%e : .--o
%e :
%e : 3: [ 0 1 2 2 1 2 2 ] [ 2 2 . . 2 . . ]
%e : O--o--o
%e : .--o
%e : .--o--o
%e : .--o
%e :
%e : 4: [ 0 1 2 2 1 1 1 ] [ 4 2 . . . . . ]
%e : O--o--o
%e : .--o
%e : .--o
%e : .--o
%e : .--o
%e :
%e : 5: [ 0 1 1 1 1 1 1 ] [ 6 . . . . . . ]
%e : O--o
%e : .--o
%e : .--o
%e : .--o
%e : .--o
%e : .--o
%e :
%e (End)
%t a[1] = 1;
%t 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)]}];
%t 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 *)
%o (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)}
%Y Cf. A115593.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Oct 26 2011
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