A000242 3rd power of rooted tree enumerator; number of linear forests of 3 rooted trees.

1, 3, 9, 25, 69, 186, 503, 1353, 3651, 9865, 26748, 72729, 198447, 543159, 1491402, 4107152, 11342826, 31408719, 87189987, 242603970, 676524372, 1890436117, 5292722721, 14845095153, 41708679697, 117372283086, 330795842217

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

References

• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
• 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

• T. D. Noe, Table of n, a(n) for n=3..200
• Index entries for sequences related to rooted trees

Crossrefs

Column 3 of A339067.

Cf. A000081, A000106, A000300, A000343, A000395.

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-2)^3, x=0, n+1), x,n): seq(a(n), n=3..29); # Alois P. Heinz, Aug 21 2008

Mathematica

max = 29; 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]}]; f[x_] := Sum[ b[k]*x^k, {k, 0, max}]; Drop[ CoefficientList[ Series[f[x]^3, {x, 0, max}], x], 3] (* Jean-François Alcover, Oct 25 2011, after Alois P. Heinz *)

Formula

G.f.: B(x)^3 where B(x) is g.f. of A000081.

a(n) ~ 3 * A187770 * A051491^n / n^(3/2). - Vaclav Kotesovec, Jan 03 2021

Extensions

More terms from Christian G. Bower, Nov 15 1999