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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)).

%I A320221 #16 Dec 09 2020 15:55:06
%S A320221 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,
%T A320221 46,1,40,70,1,54,127,1,1,65,189,1,1,87,320,5,1,104,476,10,1,136,771,
%U A320221 32,1,164,1145,63,1,209,1795,154,1,252,2657,304,1,319,4091,656
%N 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)).
%H A320221 Andrew Howroyd, <a href="/A320221/b320221.txt">Table of n, a(n) for n = 1..1154</a> (rows 1..200)
%e A320221 Triangle begins:
%e A320221   1
%e A320221   1
%e A320221   1
%e A320221   1  1
%e A320221   1  1
%e A320221   1  3
%e A320221   1  3
%e A320221   1  6  1
%e A320221   1  7  1
%e A320221   1 11  4
%e A320221   1 13  6
%e A320221   1 20 16
%e A320221   1 23 23
%e A320221   1 33 46
%e A320221   1 40 70
%e A320221 The T(11,3) = 6 rooted trees:
%e A320221    (((oo)(oo))((oo)(ooooo)))
%e A320221    (((oo)(oo))((ooo)(oooo)))
%e A320221    (((oo)(ooo))((oo)(oooo)))
%e A320221    (((oo)(ooo))((ooo)(ooo)))
%e A320221   (((oo)(oo))((oo)(oo)(ooo)))
%e A320221   (((oo)(ooo))((oo)(oo)(oo)))
%t A320221 qurt[n_]:=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[qurt/@ptn]],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}]];
%t A320221 DeleteCases[Table[Length[Select[qurt[n],SameQ[##,k]&@@Length/@Position[#,{}]&]],{n,10},{k,0,n-1}],0,{2}]
%o A320221 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A320221 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]}
%o A320221 { my(A=T(15)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Dec 09 2020
%Y A320221 Row sums are A120803. Second column is A083751. A regular version is A320179.
%Y A320221 Cf. A000669, A005804, A048816, A119262, A120803, A141268, A244925, A319312.
%Y A320221 Cf. A316624, A320154, A320155, A320160, A320172, A320173.
%K A320221 nonn,tabf
%O A320221 1,9
%A A320221 _Gus Wiseman_, Oct 07 2018
%E A320221 Terms a(36) and beyond from _Andrew Howroyd_, Dec 09 2020
%E A320221 Name clarified by _Andrew Howroyd_, Dec 09 2020