A000485 Number of partially labeled trees with n nodes (4 of which are labeled). (Formerly M5008 N2156)

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

Offset

4,1

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

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.

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 *)

Crossrefs

Column k=4 of A034799.

Sequence in context: A126511 A231582 A067442 * A264625 A213748 A007787

Adjacent sequences: A000482 A000483 A000484 * A000486 A000487 A000488

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Oct 19 2001

Status

approved