A238372 Number of labeled rooted trees with n nodes with every leaf at the same height.

1, 2, 9, 40, 265, 1956, 18529, 183520, 2226753, 28663300, 421589641, 6696832704, 117283627201, 2190260755060, 44645172510345, 964646320357696, 22317294448547329, 547594529028427908, 14246751684203363593, 390309056795283743200, 11276891642831796476481

Offset: 1

Author

F. Chapoton, Feb 25 2014

Keywords

• nonn

Examples

			On 4 vertices, there are:
24 rooted trees X-O-O-O
12 rooted trees X-O-O
                   \
                    O
4 rooted trees    X
                 /|\
                O O O
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A048816 for the unlabeled version.

Programs

Maple

p:= proc(i) p(i):= `if`(i=1, x, x*(exp(p(i-1))-1)) end:
s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+p(n)) end:
a:= n-> n! * coeff(series(s(n), x, n+1), x, n):
seq(a(n), n=1..25);  # Alois P. Heinz, Feb 26 2014

Mathematica

T[n_, n_] = 1; T[n_, m_] := T[n, m] = n!/(n-m)!*Sum[StirlingS2[k, m]*T[n-m, k], {k, 1, n-m}]; a[n_] := T[n, 1]; Array[a, 25] (* Jean-François Alcover, Jan 08 2016, after Vladimir Kruchinin *)

Maxima

T(n,m):=if n=m then 1 else n!/(n-m)!*sum(stirling2(k,m)*T(n-m,k),k,1,n-m);
makelist(T(n,1),n,1,15); /* Vladimir Kruchinin, Apr 01 2015 */

Sage

x = QQ[['x']].gen()
P = {}
N = 20
P[1] = x.O(N)
for i in range(2, N):
    P[i] = x*(P[i-1].exp(N)-1)
add(P[u] for u in P)

Formula

E.g.f.: Sum_{i>=1} P_i with P_1 = x and P_i = x * (exp(P_{i-1})-1) for i>1.

a(n) = T(n,1), T(n,m) = n!/(n-m)!*Sum_{k=1..n-m}(stirling2(k,m)*T(n-m,k)), T(n,n)=1. - Vladimir Kruchinin, Apr 01 2015