A006871 - OEIS

%I #21 Mar 18 2015 20:50:33

%S 1,2,6,22,95,470,2618,16159,109177,799959,6309111,53230208,477941835,

%T 4546562149,45646689381,482052312792,5338922526947,61851205026759,

%U 747755335385701,9413952961001366,123184994320876476,1672494478592315483,23523730530581660635

%N Exponentiation of g.f. for rooted trees.

%H Alois P. Heinz, <a href="/A006871/b006871.txt">Table of n, a(n) for n = 0..500</a>

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

%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; unapply(add(b(k)*x^k/k!, k=1..n),x) end: a:= n-> coeff(series(exp(B(n+1)(x)), x=0, n+2), x,n+1)*(n+1)!: seq(a(n), n=0..20); # _Alois P. Heinz_, Aug 22 2008

%p # second Maple program:

%p with(numtheory):

%p g:= proc(n) option remember; `if`(n<=1, n, (add(add(d*

%p       g(d), d=divisors(j))*g(n-j), j=1..n-1))/(n-1))

%p     end:

%p b:= proc(n) option remember; `if`(n=0, 1, add(

%p       binomial(n-1, j-1)*g(j)*b(n-j), j=1..n))

%p     end:

%p a:= n-> b(n+1):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 18 2015

%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] = Function[x, Evaluate @ Sum[b[k]*x^k/k!, {k, 1, n}]]; a[n_] := Coefficient[Series[Exp[B[n+1][x]], {x, 0, n+2}], x, n+1]*(n+1)!; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 17 2014, after _Alois P. Heinz_ *)

%Y Cf. A000081.

%K nonn

%O 0,2

%A _N. J. A. Sloane_.