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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.
16
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, 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, 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
OFFSET
1,14
LINKS
EXAMPLE
: T(6,4) = 3 : T(11,3) = 1 :
: o o o : o :
: / \ | | : /( )\ :
: o o o o : o o o o :
: | / \ | : /| | | :
: o o o o : o o o o :
: | | / \ : | | :
: o o o o : o o :
: | | | : :
: o o o : :
Triangle T(n,k) begins:
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, 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, 27, 8, 1;
0, 0, 0, 0, 32, 108, 149, 132, 82, 35, 9, 1;
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k<1, 0,
add(binomial(b((i-1)$2, k-1), j)*b(n-i*j, i-1, k), j=0..n/i)))
end:
T:= (n, k)-> b((n-1)$2, k) -`if`(k=0, 0, b((n-1)$2, k-1)):
seq(seq(T(n, k), k=0..n-1), n=1..15);
MATHEMATICA
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 *)
CROSSREFS
Columns k=4-10 give: A038088, A038089, A038090, A038091, A038092, A229403, A229404.
Row sums give: A004111.
Column sums give: A038081.
Largest n with T(n,k)>0 is A038093(k).
Main diagonal and lower diagonals give (offsets may differ): A000012, A001477, A000096, A166830.
T(2n,n) gives A245090.
T(2n+1,n) gives A245091.
Cf. A034781.
Sequence in context: A106278 A339829 A177517 * A064287 A196389 A128206
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 31 2013
STATUS
approved