A000526 Number of partially labeled trees with n nodes (5 of which are labeled). (Formerly M5387 N2340)

125, 1296, 8716, 47787, 232154, 1040014, 4395772, 17781210, 69498964, 264248924, 982218072, 3582421612, 12857819052, 45515994861, 159205157535, 551049504784, 1889714853263, 6427147635062, 21698583468717

Offset

5,1

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 = 5..200

Index entries for sequences related to trees

Formula

G.f.: A(x) = B(x)^5*(125-204*B(x)+118*B(x)^2-24*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-4)^5* (125-204*B(n-4) +118*B(n-4)^2 -24*B(n-4)^3)/ (1-B(n-4))^7, x=0, n+1), x, n): seq(a(n), n=5..23); # 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-4]^5*(125 - 204*B[n-4] + 118*B[n-4]^2 - 24*B[n-4]^3)/(1 - B[n-4])^7, {x, 0, n}]; Table[a[n], {n, 5, 23}] (* Jean-François Alcover, Mar 20 2014, after Alois P. Heinz *)

Crossrefs

Column k=5 of A034799.

Sequence in context: A352161 A243240 A237713 * A016971 A030082 A342349

Adjacent sequences: A000523 A000524 A000525 * A000527 A000528 A000529

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Oct 19 2001

Status

approved