A000525 Number of partially labeled rooted trees with n nodes (4 of which are labeled). (Formerly M5329 N2317)

64, 625, 4016, 21256, 100407, 439646, 1823298, 7258228, 27983518, 105146732, 386812476, 1398023732, 4977320988, 17492710572, 60790051789, 209179971147, 713533304668, 2415061934763, 8117293752058, 27111950991825, 90039381031273

Offset

4,1

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

Vincenzo Librandi, Table of n, a(n) for n = 4..200

Index entries for sequences related to rooted trees

Index entries for sequences related to trees

Formula

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.

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* (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

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*(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 *)

Crossrefs

Column k=4 of A008295.

Cf. A042977.

Sequence in context: A200788 A250355 A045789 * A067476 A179810 A303265

Adjacent sequences: A000522 A000523 A000524 * A000526 A000527 A000528

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Oct 19 2001

Status

approved