A121516 Number of 3-decomposable trees on 3n nodes.

2, 10, 84, 788, 8188, 90110, 1035456, 12269932, 148886048, 1840585914, 23099713808, 293535000452, 3769200628592, 48831588116862, 637501117219024, 8378367468484212, 110760388293651950, 1471854299855109782, 19649723961974718686, 263422552838889748560

Offset: 1

Author

N. J. A. Sloane, Sep 12 2006

Keywords

• nonn
• easy

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..100
• S. G. Wagner, An identity for the cycle indices of rooted tree automorphism groups, Elec. J. Combinat., 13 (2006), #R00.

Crossrefs

Cf. A000151.

Programs

Maple

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

Mathematica

terms = 20; A[_] = 0;
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];
DeleteCases[CoefficientList[A[x], x], 0] (* Jean-François Alcover, Apr 07 2020 *)

Formula

Wagner gives a g.f.

a(n) ~ c * d^n / n^(3/2), where d = 14.47726020066578... and c = 0.144218531921... - Vaclav Kotesovec, Apr 07 2020

Extensions

More terms from R. J. Mathar, Sep 17 2006