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A214571 Irregular triangle read by rows: T(n,k) is the number of ordered trees having n vertices and isomorphic (as rooted trees) to k ordered trees (n >= 1, k >= 1).

%I A214571 #37 Oct 13 2017 05:13:53
%S A214571 1,1,2,3,2,5,6,3,6,16,12,8,10,34,21,32,5,30,11,68,48,100,15,108,0,24,
%T A214571 9,10,0,36,16,126,81,260,30,336,7,128,45,40,0,264,0,0,15,0,0,18,0,40,
%U A214571 0,0,0,24,19,222,141,616,60,828,21,520,117,130,0,1140,0,0,45,80,0,234,0,160,21,0,0,312,0,0,0,0,0,120,0,0,0,0,0,36,0,0,0,40,26,372,219,1392,90,1914,42,1664,315,320,0,3696,0,28,195,544,0,1044,0,580,21,0,0,2112,0,0,27,28,0,480,0,0,0,0,35,648,0,0,0,320,0,84,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,0,360
%N A214571 Irregular triangle read by rows: T(n,k) is the number of ordered trees having n vertices and isomorphic (as rooted trees) to k ordered trees (n >= 1, k >= 1).
%C A214571 Row n contains A214570(n) entries.
%C A214571 T(n,1) = A003238(n).
%C A214571 Sum_{k=1..n} T(n,k) = A000108(n) (the Catalan numbers).
%C A214571 Sum_{k=1..n} T(n,k)/k = A000081(n) (the number of rooted trees with n vertices).
%C A214571 T(n,k) = k*A214569(n,k).
%C A214571 T(n,k) is also the number of function representations as x^x^...^x with n x's and parentheses inserted in all possible ways that are equivalent to (describe the same function as) k-1 other representations. T(4,2) = 2: (x^x)^(x^x), (x^(x^x))^x; T(5,3) = 3: ((x^x)^x)^(x^x), ((x^x)^(x^x))^x, ((x^(x^x))^x)^x. - _Alois P. Heinz_, Aug 31 2012
%H A214571 Alois P. Heinz, <a href="/A214571/b214571.txt">Rows n = 1..16, flattened</a>
%F A214571 No formula available. Entries have been obtained from T(n,k)= k*A214569(n,k).
%e A214571 Row 4 is 3,2: among the five ordered trees with 4 vertices the path tree P_4, the star tree K_{1,3}, and the tree in the shape of Y are isomorphic only to themselves, while A - B - C - D with root at B and A - B - C - D with root at C are isomorphic among themselves.
%e A214571 Triangle starts:
%e A214571    1;
%e A214571    1;
%e A214571    2;
%e A214571    3,   2;
%e A214571    5,   6,   3;
%e A214571    6,  16,  12,   8;
%e A214571   10,  34,  21,  32,   5,  30;
%e A214571   11,  68,  48, 100,  15, 108,   0,  24,   9,  10,   0,  36;
%p A214571 F:= proc(n) option remember; `if`(n=1, [x+1],
%p A214571       [seq(seq(seq(f^g, g=F(n-i)), f=F(i)), i=1..n-1)])
%p A214571     end:
%p A214571 T:= proc(n) option remember; local i, l, p;
%p A214571       l:= map(f->coeff(series(f, x, n+1), x, n), F(n)):
%p A214571       p:= proc() 0 end: forget(p);
%p A214571       for i in l do p(i):= p(i)+1 od:
%p A214571       l:= map(p, l); forget(p);
%p A214571       for i in l do p(i):= p(i)+1 od:
%p A214571       seq(p(i), i=1..max(l[]))
%p A214571     end:
%p A214571 seq(T(n), n=1..10);  # _Alois P. Heinz_, Aug 31 2012
%t A214571 F[n_] := F[n] = If[n == 1, {x+1}, Flatten[Table[Table[Table[f^g, {g, F[n-i]}], {f, F[i]}], {i, 1, n-1}]]]; T[n_] := T[n] = Module[{i, l, p}, l = Map[Function[ {f}, Coefficient[Series[f, {x, 0, n+1}], x, n]], F[n]]; Clear[p]; p[_] = 0; Do[ p[i] = p[i]+1 , {i, l}]; l = Map[p, l]; Clear[p]; p[_] = 0; Do[p[i] = p[i]+1, {i, l}]; Table[p[i], {i, 1, Max[l]}]]; Table[T[n], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, May 28 2015, after _Alois P. Heinz_ *)
%Y A214571 Cf. A000108, A000081, A003238, A214569, A214570, A061775, A206487, A215703.
%K A214571 nonn,tabf,hard
%O A214571 1,3
%A A214571 _Emeric Deutsch_, Jul 28 2012