A000444 Number of partially labeled rooted trees with n nodes (3 of which are labeled). (Formerly M4641 N1984)

9, 64, 326, 1433, 5799, 22224, 81987, 293987, 1031298, 3555085, 12081775, 40576240, 134919788, 444805274, 1455645411, 4733022100, 15302145060, 49223709597, 157629612076, 502736717207, 1597541346522, 5059625685739, 15975936032821, 50304490599602

Offset

3,1

References

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 134.

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

Alois P. Heinz, Table of n, a(n) for n = 3..650

Index entries for sequences related to rooted trees

Index entries for sequences related to trees

Formula

G.f.: A(x) = B(x)^3*(9-8*B(x)+2*B(x)^2)/(1-B(x))^5, where B(x) is g.f. for rooted trees with n nodes, cf. A000081.

a(n) ~ c * d^n * n^(3/2), where d = A051491 = 2.9557652856519949747148..., c = 0.244665117500618173509... . - Vaclav Kotesovec, Sep 11 2014

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-2)^3*(9-8*B(n-2)+2*B(n-2)^2)/(1-B(n-2))^5, x=0, n+1), x, n): seq(a(n), n=3..24); # 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_] := Coefficient[Series[B[n-2]^3*(9 - 8*B[n-2] + 2*B[n-2]^2)/(1 - B[n-2])^5, {x, 0, n+1}], x, n]; Table[a[n], {n, 3, 30}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)

Crossrefs

Column k=3 of A008295.

Cf. A042977.

Sequence in context: A099761 A018201 A181888 * A143631 A083328 A000846

Adjacent sequences: A000441 A000442 A000443 * A000445 A000446 A000447

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Oct 19 2001

Status

approved