A000343 5th power of rooted tree enumerator; number of linear forests of 5 rooted trees. (Formerly M3901 N1601)

1, 5, 20, 70, 230, 721, 2200, 6575, 19385, 56575, 163952, 472645, 1357550, 3888820, 11119325, 31753269, 90603650, 258401245, 736796675, 2100818555, 5990757124, 17087376630, 48753542665, 139155765455, 397356692275, 1135163887190, 3244482184720, 9277856948255

Offset

5,2

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

Alois P. Heinz, Table of n, a(n) for n = 5..750

Index entries for sequences related to rooted trees

Formula

G.f.: B(x)^5 where B(x) is g.f. of A000081.

a(n) ~ 5 * 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-4)^5, x=0, n+1), x, n): seq(a(n), n=5..29); # 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-4]^5, {x, 0, n+1}], x, n]; Table[a[n], {n, 5, 32}] (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz *)

Crossrefs

Column 5 of A339067.

Cf. A000081, A000106, A000242, A000300, A000395.

Sequence in context: A089094 A080930 A169792 * A005324 A304011 A243869

Adjacent sequences: A000340 A000341 A000342 * A000344 A000345 A000346

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Christian G. Bower, Nov 15 1999

Status

approved