%I #34 Sep 07 2018 22:08:22
%S 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 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 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 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 Alois P. Heinz, <a href="/A227819/b227819.txt">Rows n = 1..141, flattened</a>
%e : T(6,4) = 3 : T(11,3) = 1 :
%e : o o o : o :
%e : / \ | | : /( )\ :
%e : o o o o : o o o o :
%e : | / \ | : /| | | :
%e : o o o o : o o o o :
%e : | | / \ : | | :
%e : o o o o : o o :
%e : | | | : :
%e : o o o : :
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 0, 1;
%e 0, 0, 1, 1;
%e 0, 0, 0, 2, 1;
%e 0, 0, 0, 2, 3, 1;
%e 0, 0, 0, 2, 5, 4, 1;
%e 0, 0, 0, 2, 8, 9, 5, 1;
%e 0, 0, 0, 1, 12, 18, 14, 6, 1;
%e 0, 0, 0, 1, 17, 34, 33, 20, 7, 1;
%e 0, 0, 0, 1, 23, 61, 72, 54, 27, 8, 1;
%e 0, 0, 0, 0, 32, 108, 149, 132, 82, 35, 9, 1;
%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k<1, 0,
%p add(binomial(b((i-1)$2, k-1), j)*b(n-i*j, i-1, k), j=0..n/i)))
%p end:
%p T:= (n, k)-> b((n-1)$2, k) -`if`(k=0, 0, b((n-1)$2, k-1)):
%p seq(seq(T(n, k), k=0..n-1), n=1..15);
%t 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 Columns k=4-10 give: A038088, A038089, A038090, A038091, A038092, A229403, A229404.
%Y Row sums give: A004111.
%Y Column sums give: A038081.
%Y Largest n with T(n,k)>0 is A038093(k).
%Y Main diagonal and lower diagonals give (offsets may differ): A000012, A001477, A000096, A166830.
%Y T(2n,n) gives A245090.
%Y T(2n+1,n) gives A245091.
%Y Cf. A034781.
%K nonn,tabl
%O 1,14
%A _Alois P. Heinz_, Jul 31 2013