A000242 3rd power of rooted tree enumerator; number of linear forests of 3 rooted trees. (Formerly M2798 N1126)

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

3,2

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

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

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 *)

Crossrefs

Column 3 of A339067.

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

Sequence in context: A338726 A323362 A201533 * A077846 A005322 A103780

Adjacent sequences: A000239 A000240 A000241 * A000243 A000244 A000245

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Christian G. Bower, Nov 15 1999

Status

approved