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%I A320179 #12 Dec 09 2020 15:55:18
%S A320179 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,
%T A320179 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,
%U A320179 1,20,16,0,0,0,0,0,0,0,0,0,1,23,23,0,0,0
%N 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.
%H A320179 Andrew Howroyd, <a href="/A320179/b320179.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%e A320179 Triangle begins:
%e A320179   1
%e A320179   0  1
%e A320179   0  1  0
%e A320179   0  1  1  0
%e A320179   0  1  1  0  0
%e A320179   0  1  3  0  0  0
%e A320179   0  1  3  0  0  0  0
%e A320179   0  1  6  1  0  0  0  0
%e A320179   0  1  7  1  0  0  0  0  0
%e A320179   0  1 11  4  0  0  0  0  0  0
%e A320179   0  1 13  6  0  0  0  0  0  0  0
%e A320179   0  1 20 16  0  0  0  0  0  0  0  0
%e A320179   0  1 23 23  0  0  0  0  0  0  0  0  0
%e A320179   0  1 33 46  0  0  0  0  0  0  0  0  0  0
%e A320179 The T(10,3) = 4 rooted trees:
%e A320179    (((oo)(oo))((oo)(oooo)))
%e A320179    (((oo)(oo))((ooo)(ooo)))
%e A320179    (((oo)(ooo))((oo)(ooo)))
%e A320179   (((oo)(oo))((oo)(oo)(oo)))
%t A320179 qurt[n_]:=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[qurt/@ptn]],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}]];
%t A320179 Table[Length[Select[qurt[n],SameQ[##,k]&@@Length/@Position[#,{}]&]],{n,14},{k,0,n-1}]
%o A320179 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A320179 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))}
%o A320179 { my(A=T(15)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Dec 09 2020
%Y A320179 Row sums are A120803. Third column is A083751. An irregular version is A320221.
%Y A320179 Cf. A000669, A001678, A048816, A079500, A119262, A244925, A316655, A319312.
%Y A320179 Cf. A316624, A320154, A320155, A320160, A320172, A320173.
%K A320179 nonn,tabl
%O A320179 1,18
%A A320179 _Gus Wiseman_, Oct 07 2018

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