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A320221
Irregular 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, (n>=1, min(1,n-1) <= k <= log_2(n)).
2
1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 6, 1, 1, 7, 1, 1, 11, 4, 1, 13, 6, 1, 20, 16, 1, 23, 23, 1, 33, 46, 1, 40, 70, 1, 54, 127, 1, 1, 65, 189, 1, 1, 87, 320, 5, 1, 104, 476, 10, 1, 136, 771, 32, 1, 164, 1145, 63, 1, 209, 1795, 154, 1, 252, 2657, 304, 1, 319, 4091, 656
OFFSET
1,9
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1154 (rows 1..200)
EXAMPLE
Triangle begins:
1
1
1
1 1
1 1
1 3
1 3
1 6 1
1 7 1
1 11 4
1 13 6
1 20 16
1 23 23
1 33 46
1 40 70
The T(11,3) = 6 rooted trees:
(((oo)(oo))((oo)(ooooo)))
(((oo)(oo))((ooo)(oooo)))
(((oo)(ooo))((oo)(oooo)))
(((oo)(ooo))((ooo)(ooo)))
(((oo)(oo))((oo)(oo)(ooo)))
(((oo)(ooo))((oo)(oo)(oo)))
MATHEMATICA
qurt[n_]:=If[n==1, {{}}, Join@@Table[Union[Sort/@Tuples[qurt/@ptn]], {ptn, Select[IntegerPartitions[n], Length[#]>1&]}]];
DeleteCases[Table[Length[Select[qurt[n], SameQ[##, k]&@@Length/@Position[#, {}]&]], {n, 10}, {k, 0, n-1}], 0, {2}]
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); v[1]=x; [Vecrev(p/x) | p<-v]}
{ my(A=T(15)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Dec 09 2020
CROSSREFS
Row sums are A120803. Second column is A083751. A regular version is A320179.
Sequence in context: A146907 A322865 A238694 * A236939 A236936 A236915
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Oct 07 2018
EXTENSIONS
Terms a(36) and beyond from Andrew Howroyd, Dec 09 2020
Name clarified by Andrew Howroyd, Dec 09 2020
STATUS
approved