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A320179
Regular triangle where T(n,k) is the number of unlabeled series-reduced rooted trees with n leaves in which every leaf is at height k.
9
1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 3, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 6, 1, 0, 0, 0, 0, 0, 1, 7, 1, 0, 0, 0, 0, 0, 0, 1, 11, 4, 0, 0, 0, 0, 0, 0, 0, 1, 13, 6, 0, 0, 0, 0, 0, 0, 0, 0, 1, 20, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 23, 23, 0, 0, 0
OFFSET
1,18
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
EXAMPLE
Triangle begins:
1
0 1
0 1 0
0 1 1 0
0 1 1 0 0
0 1 3 0 0 0
0 1 3 0 0 0 0
0 1 6 1 0 0 0 0
0 1 7 1 0 0 0 0 0
0 1 11 4 0 0 0 0 0 0
0 1 13 6 0 0 0 0 0 0 0
0 1 20 16 0 0 0 0 0 0 0 0
0 1 23 23 0 0 0 0 0 0 0 0 0
0 1 33 46 0 0 0 0 0 0 0 0 0 0
The T(10,3) = 4 rooted trees:
(((oo)(oo))((oo)(oooo)))
(((oo)(oo))((ooo)(ooo)))
(((oo)(ooo))((oo)(ooo)))
(((oo)(oo))((oo)(oo)(oo)))
MATHEMATICA
qurt[n_]:=If[n==1, {{}}, Join@@Table[Union[Sort/@Tuples[qurt/@ptn]], {ptn, Select[IntegerPartitions[n], Length[#]>1&]}]];
Table[Length[Select[qurt[n], SameQ[##, k]&@@Length/@Position[#, {}]&]], {n, 14}, {k, 0, n-1}]
PROG
(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
T(n)={my(u=vector(n), v=vector(n), h=1); u[1]=1; while(u, v+=u*h; h*=x; u=EulerT(u)-u); vector(n, n, Vecrev(v[n], n))}
{ my(A=T(15)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Dec 09 2020
CROSSREFS
Row sums are A120803. Third column is A083751. An irregular version is A320221.
Sequence in context: A111417 A007271 A327449 * A035656 A325675 A374203
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Oct 07 2018
STATUS
approved