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%I #11 Dec 01 2023 05:12:35
%S 0,1,1,4,9,22,54,138,346,889,2285,5928,15436,40424,106230,280305,
%T 741912,1969816,5243942,13995807,37439883,100371907,269623436,
%U 725638613,1956352468,5283171593,14289645110,38707131195,104995130162,285184002486,775586517781
%N Number of rooted trees with n nodes such that two equals the maximal number of isomorphic subtrees extending from the same node.
%H Alois P. Heinz, <a href="/A318859/b318859.txt">Table of n, a(n) for n = 2..2213</a>
%p h:= proc(n, m, t, k) option remember; `if`(m=0, binomial(n+t, t),
%p `if`(n=0, 0, add(h(n-1, m-j, t+1, k), j=1..min(k, m))))
%p end:
%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(b(n-i*j, i-1, k)*h(A(i, k), j, 0, k), j=0..n/i)))
%p end:
%p A:= (n, k)-> `if`(n<2, n, b(n-1$2, k)):
%p a:= n-> (k-> A(n, k)-A(n, k-1))(2):
%p seq(a(n), n=2..32);
%t h[n_, m_, t_, k_] := h[n, m, t, k] = If[m == 0, Binomial[n + t, t],
%t If[n == 0, 0, Sum[h[n - 1, m - j, t + 1, k], {j, 1, Min[k, m]}]]];
%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,
%t Sum[b[n - i*j, i - 1, k]*h[A[i, k], j, 0, k], {j, 0, n/i}]]];
%t A[n_, k_] := If[n < 2, n, b[n - 1, n - 1, k]];
%t a[n_] := A[n, 2] - A[n, 1];
%t Table[a[n], {n, 2, 32}] (* _Jean-François Alcover_, Dec 01 2023, after _Alois P. Heinz_ *)
%Y Column k=2 of A318758.
%K nonn
%O 2,4
%A _Alois P. Heinz_, Sep 04 2018
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