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A242375 Number of rooted trees with n n-colored non-root nodes.

%I A242375 #18 Mar 18 2024 05:57:06
%S A242375 1,1,7,82,1499,37476,1200705,46990952,2175619923,116400215521,
%T A242375 7069820334023,480722969498938,36186340018129392,2987845924408179654,
%U A242375 268530017303221572650,26098422892000807053155,2727654868575748827350403,305075571192329680642519141
%N A242375 Number of rooted trees with n n-colored non-root nodes.
%H A242375 Alois P. Heinz, <a href="/A242375/b242375.txt">Table of n, a(n) for n = 0..200</a>
%F A242375 a(n) ~ c * exp(n) * n^(n-3/2), where c = exp(1 + exp(-2)/2) / sqrt(2*Pi) = 1.160358615244339554387715748... . - _Vaclav Kotesovec_, Aug 28 2014, updated Mar 18 2024
%e A242375 a(2) = 7:
%e A242375   o    o    o    o      o        o        o
%e A242375   |    |    |    |     / \      / \      / \
%e A242375   1    1    2    2    1   1    1   2    2   2
%e A242375   |    |    |    |
%e A242375   1    2    1    2
%p A242375 with(numtheory):
%p A242375 b:= proc(n, k) option remember; `if`(n<2, n, (add(add(d*
%p A242375       b(d, k), d=divisors(j))*b(n-j, k)*k, j=1..n-1))/(n-1))
%p A242375     end:
%p A242375 a:= n-> b(n+1, n):
%p A242375 seq(a(n), n=0..20);
%t A242375 b[n_, k_] := b[n, k] = If[n < 2, n, (Sum[Sum[d*b[d, k], {d, Divisors[j]}] * b[n - j, k]*k, {j, 1, n - 1}])/(n - 1)];
%t A242375 a[n_] := b[n + 1, n];
%t A242375 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 21 2017, translated from Maple *)
%Y A242375 A diagonal of A242249.
%Y A242375 Cf. A255523.
%K A242375 nonn
%O A242375 0,3
%A A242375 _Alois P. Heinz_, May 12 2014