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Number of series-reduced anti-binary (no unary or binary branchings) unlabeled rooted trees with n nodes.
6

%I #19 May 17 2021 15:05:08

%S 1,0,0,1,1,1,2,3,4,7,11,17,28,46,74,123,205,341,571,964,1629,2764,

%T 4707,8040,13766,23639,40681,70163,121256,209960,364168,632694,

%U 1100906,1918375,3347346,5848271,10229977,17915018,31407088,55116661,96818589,170229939

%N Number of series-reduced anti-binary (no unary or binary branchings) unlabeled rooted trees with n nodes.

%H Alois P. Heinz, <a href="/A303025/b303025.txt">Table of n, a(n) for n = 1..2000</a>

%e The a(10) = 7 rooted trees:

%e (oo(oo(ooo)))

%e (o(ooo)(ooo))

%e (oo(oooooo))

%e (ooo(ooooo))

%e (oooo(oooo))

%e (ooooo(ooo))

%e (ooooooooo)

%p b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0), `if`(i<1, 0,

%p add(b(n-i*j, i-1, max(0, t-j))*binomial(a(i)+j-1, j), j=0..n/i)))

%p end:

%p a:= n-> `if`(n<2, n, b(n-1$2, 3)):

%p seq(a(n), n=1..50); # _Alois P. Heinz_, Aug 27 2018

%t zurt[n_]:=zurt[n]=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[zurt/@c]],{c,Select[IntegerPartitions[n-1],Length[#]>2&]}]];

%t Table[Length[zurt[n]],{n,20}]

%t (* Second program: *)

%t b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], If[i < 1, 0, Sum[b[n-i*j, i - 1, Max[0, t-j]]*Binomial[a[i]+j-1, j], {j, 0, n/i}]]];

%t a[n_] := If[n < 2, n, b[n-1, n-1, 3]];

%t Array[a, 50] (* _Jean-François Alcover_, May 17 2021, after _Alois P. Heinz_ *)

%Y Cf. A000081, A000598, A001190, A001678, A102403, A298204, A298422.

%Y Cf. A303022, A303023, A303024, A303026, A303027.

%K nonn

%O 1,7

%A _Gus Wiseman_, Aug 15 2018

%E a(36)-a(42) from _Alois P. Heinz_, Aug 27 2018