OFFSET
1,11
COMMENTS
Row sums give A004111.
LINKS
Joerg Arndt and Alois P. Heinz, Rows n = 1..285, flattened
EXAMPLE
Triangle starts:
01: 1,
02: 0, 1,
03: 0, 1,
04: 0, 1, 1,
05: 0, 1, 2,
06: 0, 1, 5,
07: 0, 1, 10, 1,
08: 0, 1, 21, 3,
09: 0, 1, 42, 9,
10: 0, 1, 87, 25,
11: 0, 1, 178, 66, 2,
12: 0, 1, 371, 170, 6,
13: 0, 1, 773, 431, 21,
14: 0, 1, 1630, 1076, 63,
15: 0, 1, 3447, 2665, 185, 1,
16: 0, 1, 7346, 6560, 512, 7,
17: 0, 1, 15712, 16067, 1403, 26,
18: 0, 1, 33790, 39219, 3750, 91,
19: 0, 1, 72922, 95476, 9928, 291,
20: 0, 1, 158020, 231970, 25969, 885, 3,
21: 0, 1, 343494, 562736, 67462, 2588, 15,
22: 0, 1, 749101, 1363640, 174039, 7373, 70,
23: 0, 1, 1638102, 3301586, 446884, 20555, 256,
24: 0, 1, 3591723, 7988916, 1142457, 56413, 884,
25: 0, 1, 7893801, 19322585, 2911078, 152812, 2840, 3,
...
The A004111(7) = 12 level-sequences and the branching sequences for the identity trees with 7 nodes are (dots for zeros), together with the maximal branching factors, are:
01: [ . 1 2 3 4 5 6 ] [ 1 1 1 1 1 1 . ] 1
02: [ . 1 2 3 4 5 4 ] [ 1 1 1 2 1 . . ] 2
03: [ . 1 2 3 4 5 3 ] [ 1 1 2 1 1 . . ] 2
04: [ . 1 2 3 4 5 2 ] [ 1 2 1 1 1 . . ] 2
05: [ . 1 2 3 4 5 1 ] [ 2 1 1 1 1 . . ] 2
06: [ . 1 2 3 4 3 2 ] [ 1 2 2 1 . . . ] 2
07: [ . 1 2 3 4 3 1 ] [ 2 1 2 1 . . . ] 2
08: [ . 1 2 3 4 2 3 ] [ 1 2 1 1 . 1 . ] 2
09: [ . 1 2 3 4 2 1 ] [ 2 2 1 1 . . . ] 2
10: [ . 1 2 3 4 1 2 ] [ 2 1 1 1 . 1 . ] 2
11: [ . 1 2 3 2 1 2 ] [ 2 2 1 . . 1 . ] 2
12: [ . 1 2 3 1 2 1 ] [ 3 1 1 . 1 . . ] 3
This gives row n=7: [0, 1, 10, 1, 0, 0, ... ].
MAPLE
b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(binomial(b(i-1$2, k$2), j)*
b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
end:
g:= proc(n) local k; if n=1 then 0 else
for k while T(n, k)>0 do od; k-1 fi
end:
T:= (n, k)-> b(n-1$2, k$2) -`if`(k=0, 0, b(n-1$2, k-1$2)):
seq(seq(T(n, k), k=0..g(n)), n=1..25);
MATHEMATICA
b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[b[i-1, i-1, k, k], j]*b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]]]; g[n_] := If[ n == 1 , 0, For[k=1, T[n, k]>0 , k++]; k-1]; T[n_, k_] := b[n-1, n-1, k, k] - If[k == 0, 0, b[n-1, n-1, k-1, k-1]]; Table[Table[T[n, k], {k, 0, g[n]}], {n, 1, 25}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Maple *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Joerg Arndt and Alois P. Heinz, Jul 30 2014
STATUS
approved