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A320221
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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)).
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2
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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
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OFFSET
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1,9
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LINKS
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EXAMPLE
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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)))
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MATHEMATICA
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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}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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