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%I #35 Feb 29 2020 18:02:05
%S 1,2,9,40,265,1956,18529,183520,2226753,28663300,421589641,6696832704,
%T 117283627201,2190260755060,44645172510345,964646320357696,
%U 22317294448547329,547594529028427908,14246751684203363593,390309056795283743200,11276891642831796476481
%N Number of labeled rooted trees with n nodes with every leaf at the same height.
%H Alois P. Heinz, <a href="/A238372/b238372.txt">Table of n, a(n) for n = 1..200</a>
%F E.g.f.: Sum_{i>=1} P_i with P_1 = x and P_i = x * (exp(P_{i-1})-1) for i>1.
%F 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
%e On 4 vertices, there are:
%e 24 rooted trees X-O-O-O
%e 12 rooted trees X-O-O
%e \
%e O
%e 4 rooted trees X
%e /|\
%e O O O
%p p:= proc(i) p(i):= `if`(i=1, x, x*(exp(p(i-1))-1)) end:
%p s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+p(n)) end:
%p a:= n-> n! * coeff(series(s(n), x, n+1), x, n):
%p seq(a(n), n=1..25); # _Alois P. Heinz_, Feb 26 2014
%t 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_ *)
%o (Sage)
%o x = QQ[['x']].gen()
%o P = {}
%o N = 20
%o P[1] = x.O(N)
%o for i in range(2, N):
%o P[i] = x*(P[i-1].exp(N)-1)
%o add(P[u] for u in P)
%o (Maxima)
%o T(n,m):=if n=m then 1 else n!/(n-m)!*sum(stirling2(k,m)*T(n-m,k),k,1,n-m);
%o makelist(T(n,1),n,1,15); /* _Vladimir Kruchinin_, Apr 01 2015 */
%Y Cf. A048816 for the unlabeled version.
%K nonn
%O 1,2
%A _F. Chapoton_, Feb 25 2014
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