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%I #16 Apr 07 2020 04:20:43
%S 2,10,84,788,8188,90110,1035456,12269932,148886048,1840585914,
%T 23099713808,293535000452,3769200628592,48831588116862,
%U 637501117219024,8378367468484212,110760388293651950,1471854299855109782,19649723961974718686,263422552838889748560
%N Number of 3-decomposable trees on 3n nodes.
%H Vaclav Kotesovec, <a href="/A121516/b121516.txt">Table of n, a(n) for n = 1..100</a>
%H S. G. Wagner, <a href="http://finanz.math.tu-graz.ac.at/~wagner/identity.pdf">An identity for the cycle indices of rooted tree automorphism groups</a>, Elec. J. Combinat., 13 (2006), #R00.
%F Wagner gives a g.f.
%F a(n) ~ c * d^n / n^(3/2), where d = 14.47726020066578... and c = 0.144218531921... - _Vaclav Kotesovec_, Apr 07 2020
%p Nmax := 30 : nmax := 3*Nmax+1 : a := array(0..nmax) ; Dx := proc(z) global nmax, a ; local resul,i ; resul := 0 ; for i from 1 to (nmax+1)/3 do resul := resul+a[3*i]*z^(3*i) : od : RETURN(resul) ; end: exp1 := proc() global nmax, a ; local m,t ; t := 0 ; for m from 1 to nmax do t := t+3*Dx(x^m)/m ; od: return( taylor(exp(t),x=0,nmax+1) ) ; end: exp2 := proc() global nmax, a ; local m,t ; t := 0 ; for m from 1 to nmax do t := t+(Dx(x^m)+Dx(x^(2*m)))/m ; od: return( taylor(exp(t),x=0,nmax+1) ) ; end: DD := Dx(x)-3*x^3*exp1()/2-x^3*exp2()/2 : for i from 0 to nmax do a[i] := solve(coeftayl(DD,x=0,i),a[i]) ; if i mod 3 = 0 then print(a[i]) ; fi ; end: # _R. J. Mathar_, Sep 17 2006
%t terms = 20; A[_] = 0;
%t Do[A[x_] = (3x^3/2)Exp[Sum[(3/m)A[x^m], {m, 3 terms}]]+(x^3/2)Exp[Sum[(1/m) (A[x^m]+A[x^(2m)]), {m, 3terms}]] + O[x]^(3terms+1) // Normal, 3terms+1];
%t DeleteCases[CoefficientList[A[x], x], 0] (* _Jean-François Alcover_, Apr 07 2020 *)
%Y Cf. A000151.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_, Sep 12 2006
%E More terms from _R. J. Mathar_, Sep 17 2006
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