A064785 - OEIS

%I #22 Mar 24 2023 09:04:14

%S 1296,16807,134960,858578,4741835,23786827,111254536,493289047,

%T 2096891419,8614217489,34402073301,134162057607,512703873915,

%U 1925300176534,7120276125066,25981116938906,93678940211218

%N Number of partially labeled trees with n nodes (6 of which are labeled).

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.

%F G.f.: A(x) = B(x)^6*(1296-2633*B(x)+2128*B(x)^2-806*B(x)^3+120*B(x)^4)/(1-B(x))^9, where B(x) is g.f. for rooted trees with n nodes, cf. A000081.

%p b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n-1) fi end: s:= proc(n,k) option remember; add(b(n+1-j*k), j=1..iquo(n,k)) end: B:= proc(n) option remember; add(b(k)*x^k, k=1..n) end: a:= n-> coeff(series(B(n-2)^6* (1296-2633* B(n-2)+ 2128*B(n-2)^2 -806*B(n-2)^3 +120*B(n-2)^4)/ (1-B(n-2))^9, x=0, n+1),x,n): seq(a(n), n=6..22); # _Alois P. Heinz_, Aug 22 2008

%t jmax = 23; B[_] = 0;

%t Do[B[x_] = x*Exp[Sum[B[x^k]/k, {k, 1, j}]]+O[x]^j // Normal, {j, 1, jmax}];

%t A[x_] = B[x]^6*(1296 - 2633*B[x] + 2128*B[x]^2 - 806*B[x]^3 + 120*B[x]^4)/ (1 - B[x])^9;

%t CoefficientList[A[x] + O[x]^jmax, x] // Drop[#, 6]& (* _Jean-François Alcover_, Apr 25 2022 *)

%Y Column k=6 of A034799.

%Y Cf. A042977.

%K nonn

%O 6,1

%A _Vladeta Jovovic_, Oct 19 2001