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A227819 Number T(n,k) of n-node rooted identity trees of height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

%I A227819 #34 Sep 07 2018 22:08:22
%S A227819 1,0,1,0,0,1,0,0,1,1,0,0,0,2,1,0,0,0,2,3,1,0,0,0,2,5,4,1,0,0,0,2,8,9,
%T A227819 5,1,0,0,0,1,12,18,14,6,1,0,0,0,1,17,34,33,20,7,1,0,0,0,1,23,61,72,54,
%U A227819 27,8,1,0,0,0,0,32,108,149,132,82,35,9,1,0,0,0,0,41,187,301,303,221,118,44,10,1
%N A227819 Number T(n,k) of n-node rooted identity trees of height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.
%H A227819 Alois P. Heinz, <a href="/A227819/b227819.txt">Rows n = 1..141, flattened</a>
%e A227819 :   T(6,4) = 3              :  T(11,3) = 1  :
%e A227819 :     o       o       o     :        o      :
%e A227819 :    / \      |       |     :      /( )\    :
%e A227819 :   o   o     o       o     :     o o o o   :
%e A227819 :   |        / \      |     :    /| | |     :
%e A227819 :   o       o   o     o     :   o o o o     :
%e A227819 :   |       |        / \    :   |   |       :
%e A227819 :   o       o       o   o   :   o   o       :
%e A227819 :   |       |       |       :               :
%e A227819 :   o       o       o       :               :
%e A227819 Triangle T(n,k) begins:
%e A227819   1;
%e A227819   0, 1;
%e A227819   0, 0, 1;
%e A227819   0, 0, 1, 1;
%e A227819   0, 0, 0, 2,  1;
%e A227819   0, 0, 0, 2,  3,   1;
%e A227819   0, 0, 0, 2,  5,   4,   1;
%e A227819   0, 0, 0, 2,  8,   9,   5,   1;
%e A227819   0, 0, 0, 1, 12,  18,  14,   6,  1;
%e A227819   0, 0, 0, 1, 17,  34,  33,  20,  7,  1;
%e A227819   0, 0, 0, 1, 23,  61,  72,  54, 27,  8, 1;
%e A227819   0, 0, 0, 0, 32, 108, 149, 132, 82, 35, 9, 1;
%p A227819 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k<1, 0,
%p A227819       add(binomial(b((i-1)$2, k-1), j)*b(n-i*j, i-1, k), j=0..n/i)))
%p A227819     end:
%p A227819 T:= (n, k)-> b((n-1)$2, k) -`if`(k=0, 0, b((n-1)$2, k-1)):
%p A227819 seq(seq(T(n, k), k=0..n-1), n=1..15);
%t A227819 Drop[Transpose[Map[PadRight[#,15]&,Table[f[n_]:=Nest[ CoefficientList[ Series[ Product[(1+x^i)^#[[i]],{i,1,Length[#]}],{x,0,15}],x]&,{1},n]; f[m]-PadRight[f[m-1],Length[f[m]]],{m,1,15}]]],1]//Grid (* _Geoffrey Critzer_, Aug 01 2013 *)
%Y A227819 Columns k=4-10 give: A038088, A038089, A038090, A038091, A038092, A229403, A229404.
%Y A227819 Row sums give: A004111.
%Y A227819 Column sums give: A038081.
%Y A227819 Largest n with T(n,k)>0 is A038093(k).
%Y A227819 Main diagonal and lower diagonals give (offsets may differ): A000012, A001477, A000096, A166830.
%Y A227819 T(2n,n) gives A245090.
%Y A227819 T(2n+1,n) gives A245091.
%Y A227819 Cf. A034781.
%K A227819 nonn,tabl
%O A227819 1,14
%A A227819 _Alois P. Heinz_, Jul 31 2013