%I M5329 N2317 #41 Mar 24 2023 09:04:37
%S 64,625,4016,21256,100407,439646,1823298,7258228,27983518,105146732,
%T 386812476,1398023732,4977320988,17492710572,60790051789,209179971147,
%U 713533304668,2415061934763,8117293752058,27111950991825,90039381031273
%N Number of partially labeled rooted trees with n nodes (4 of which are labeled).
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 134.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A000525/b000525.txt">Table of n, a(n) for n = 4..200</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F G.f.: A(x) = B(x)^4*(64-79*B(x)+36*B(x)^2-6*B(x)^3)/(1-B(x))^7, 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-3)^4* (64-79*B(n-3)+ 36*B(n-3)^2- 6*B(n-3)^3)/ (1-B(n-3))^7, x=0, n+1),x,n): seq(a(n), n=4..24); # _Alois P. Heinz_, Aug 21 2008
%t b[n_] := b[n] = If[n <= 1, n, Sum[k*b[k]*s[n-1, k], {k, 1, n-1}]/(n-1)]; s[n_, k_] := s[n, k] = Sum[b[n + 1 - j*k], {j, 1, Quotient[n, k]}]; B[n_] := B[n] = Sum[b[k]*x^k, {k, 1, n}]; a[n_] := SeriesCoefficient[B[n-3]^4*(64 - 79*B[n-3] + 36*B[n-3]^2 - 6*B[n-3]^3)/ (1 - B[n-3])^7, {x, 0, n}]; Table[a[n], {n, 4, 24}] (* _Jean-François Alcover_, Mar 20 2014, after _Alois P. Heinz_ *)
%Y Column k=4 of A008295.
%Y Cf. A042977.
%K nonn
%O 4,1
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Oct 19 2001