OFFSET
1,8
COMMENTS
T(1,0) = 1 by convention.
Sum_{i=2..n-1} T(n,i) = A001678(n+1) for n>1.
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
EXAMPLE
The A000081(5) = 9 rooted trees with 5 nodes sorted by minimal outdegree of inner nodes are:
: 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 o o o o o o o o : o o : :
: | / \ | | : : :
: o o o o o : : :
: | : : :
: o : : :
: : : :
: ------------------1------------------ : ---2--- : ---4--- :
Thus row 5 = [0, 7, 1, 0, 1].
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 3, 0, 1;
0, 7, 1, 0, 1;
0, 17, 2, 0, 0, 1;
0, 42, 4, 1, 0, 0, 1;
0, 105, 7, 2, 0, 0, 0, 1;
0, 267, 15, 2, 1, 0, 0, 0, 1;
0, 684, 28, 4, 2, 0, 0, 0, 0, 1;
0, 1775, 56, 7, 2, 1, 0, 0, 0, 0, 1;
0, 4639, 110, 12, 2, 2, 0, 0, 0, 0, 0, 1;
MAPLE
b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],
1, 0), `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
b(n-i*j, i-1, max(0, t-j), k), j=0..n/i)))
end:
T:= (n, k)-> b(n-1$2, k$2) -`if`(n=1 and k=0, 0, b(n-1$2, k+1$2)):
seq(seq(T(n, k), k=0..n-1), n=1..14);
MATHEMATICA
b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[i<1, 0, Sum[Binomial[b[i-1, i-1, k, k]+j-1, j]* b[n-i*j, i-1, Max[0, t-j], k], {j, 0, n/i}]]]; T[n_, k_] := b[n-1, n-1, k, k] - If[n == 1 && k == 0, 0, b[n-1, n-1, k+1, k+1]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 08 2015, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt and Alois P. Heinz, Jun 28 2014
STATUS
approved