A320221 - OEIS

%I #16 Dec 09 2020 15:55:06

%S 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 46,1,40,70,1,54,127,1,1,65,189,1,1,87,320,5,1,104,476,10,1,136,771,

%U 32,1,164,1145,63,1,209,1795,154,1,252,2657,304,1,319,4091,656

%N 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 Andrew Howroyd, <a href="/A320221/b320221.txt">Table of n, a(n) for n = 1..1154</a> (rows 1..200)

%e Triangle begins:

%e   1

%e   1

%e   1

%e   1  1

%e   1  1

%e   1  3

%e   1  3

%e   1  6  1

%e   1  7  1

%e   1 11  4

%e   1 13  6

%e   1 20 16

%e   1 23 23

%e   1 33 46

%e   1 40 70

%e The T(11,3) = 6 rooted trees:

%e    (((oo)(oo))((oo)(ooooo)))

%e    (((oo)(oo))((ooo)(oooo)))

%e    (((oo)(ooo))((oo)(oooo)))

%e    (((oo)(ooo))((ooo)(ooo)))

%e   (((oo)(oo))((oo)(oo)(ooo)))

%e   (((oo)(ooo))((oo)(oo)(oo)))

%t qurt[n_]:=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[qurt/@ptn]],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}]];

%t DeleteCases[Table[Length[Select[qurt[n],SameQ[##,k]&@@Length/@Position[#,{}]&]],{n,10},{k,0,n-1}],0,{2}]

%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

%o 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 { my(A=T(15)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Dec 09 2020

%Y Row sums are A120803. Second column is A083751. A regular version is A320179.

%Y Cf. A000669, A005804, A048816, A119262, A120803, A141268, A244925, A319312.

%Y Cf. A316624, A320154, A320155, A320160, A320172, A320173.

%K nonn,tabf

%O 1,9

%A _Gus Wiseman_, Oct 07 2018

%E Terms a(36) and beyond from _Andrew Howroyd_, Dec 09 2020

%E Name clarified by _Andrew Howroyd_, Dec 09 2020