A000343 - OEIS

%I M3901 N1601 #33 Jan 03 2021 04:31:08

%S 1,5,20,70,230,721,2200,6575,19385,56575,163952,472645,1357550,

%T 3888820,11119325,31753269,90603650,258401245,736796675,2100818555,

%U 5990757124,17087376630,48753542665,139155765455,397356692275,1135163887190,3244482184720,9277856948255

%N 5th power of rooted tree enumerator; number of linear forests of 5 rooted trees.

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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A000343/b000343.txt">Table of n, a(n) for n = 5..750</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

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

%F a(n) ~ 5 * A187770 * A051491^n / n^(3/2). - _Vaclav Kotesovec_, Jan 03 2021

%p 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

%t 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_ *)

%Y Column 5 of A339067.

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

%K nonn

%O 5,2

%A _N. J. A. Sloane_

%E More terms from _Christian G. Bower_, Nov 15 1999