A000485 Number of partially labeled trees with n nodes (4 of which are labeled).

16, 125, 680, 3135, 13155, 51873, 195821, 715614, 2550577, 8911942, 30640888, 103951415, 348724844, 1158722880, 3818514232, 12493703403, 40620949971, 131336770375, 422536529249, 1353341880777, 4317248276746, 13722302173753

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn

References

• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 4..1000
• Index entries for sequences related to trees

Crossrefs

Column k=4 of A034799.

Programs

Maple

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*(16-19*B(n-3)+6*B(n-3)^2)/(1-B(n-3))^5, x=0, n+1), x,n): seq(a(n), n=4..25); # Alois P. Heinz, Aug 21 2008

Mathematica

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*(16-19*B[n-3] + 6*B[n-3]^2)/(1-B[n-3])^5, {x, 0, n}]; Table[a[n], {n, 4, 25}] (* Jean-François Alcover, Mar 20 2014, after Alois P. Heinz *)

Formula

G.f.: A(x) = B(x)^4*(16-19*B(x)+6*B(x)^2)/(1-B(x))^5, where B(x) is g.f. for rooted trees with n nodes, cf. A000081.

Extensions

More terms from Vladeta Jovovic, Oct 19 2001