# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a227819 Showing 1-1 of 1 %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, Rows n = 1..141, flattened %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 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE