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A124343
Number of rooted trees on n nodes with thinning limbs.
13
1, 1, 2, 3, 6, 10, 21, 38, 78, 153, 314, 632, 1313, 2700, 5646, 11786, 24831, 52348, 111027, 235834, 502986, 1074739, 2303146, 4944507, 10639201, 22930493, 49511948, 107065966, 231874164, 502834328, 1091842824, 2373565195, 5165713137, 11254029616, 24542260010
OFFSET
1,3
COMMENTS
A rooted tree with thinning limbs is such that if a node has k children, all its children have at most k children.
LINKS
EXAMPLE
The a(5) = 6 trees are ((((o)))), (o((o))), (o(oo)), ((o)(o)), (oo(o)), (oooo). - Gus Wiseman, Jan 25 2018
MAPLE
b:= proc(n, i, h, v) option remember; `if`(n=0,
`if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
`if`(n=v, 1, add(binomial(A(i, min(i-1, h))+j-1, j)
*b(n-i*j, i-1, h, v-j), j=0..min(n/i, v)))))
end:
A:= proc(n, k) option remember;
`if`(n<2, n, add(b(n-1$2, j$2), j=1..min(k, n-1)))
end:
a:= n-> A(n$2):
seq(a(n), n=1..35); # Alois P. Heinz, Jul 08 2014
MATHEMATICA
b[n_, i_, h_, v_] := b[n, i, h, v] = If[n==0, If[v==0, 1, 0], If[i<1 || v<1 || n<v, 0, If[n==v, 1, Sum[Binomial[A[i, Min[i-1, h]]+j-1, j]*b[n-i*j, i-1, h, v-j], {j, 0, Min[n/i, v]}]]]];
A[n_, k_] := A[n, k] = If[n<2, n, Sum[b[n-1, n-1, j, j], {j, 1, Min[k, n-1] }]];
a[n_] := A[n, n];
Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Mar 01 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Alois P. Heinz, Jul 04 2014
STATUS
approved