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A244704 Number of n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) 3.

%I A244704 #12 Dec 27 2020 19:44:23
%S A244704 1,1,3,6,13,25,55,107,224,454,938,1916,3969,8163,16918,35010,72724,
%T A244704 151093,314749,656115,1370348,2864948,5998547,12572884,26385837,
%U A244704 55431031,116577538,245415158,517152607,1090771973,2302729115,4865449045,10288826434,21774842539
%N A244704 Number of n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) 3.
%C A244704 In a rooted tree with thinning limbs the outdegree of a parent node is larger than or equal to the outdegree of any of its child nodes.
%H A244704 Alois P. Heinz, <a href="/A244704/b244704.txt">Table of n, a(n) for n = 4..1000</a>
%F A244704 a(n) ~ c * d^n / n^(3/2), where d = 2.1991393868..., c = 1.0259536... . - _Vaclav Kotesovec_, Aug 27 2014
%e A244704 a(7) = 6:
%e A244704     o       o        o      o        o       o
%e A244704    /|\     /|\      /|\    /|\     / | \    /|\
%e A244704   o o o   o o o    o o o  o o o   o  o  o  o o o
%e A244704   |      ( )      /|\     | |    ( ) |     | | |
%e A244704   o      o o     o o o    o o    o o o     o o o
%e A244704   |      |                |
%e A244704   o      o                o
%e A244704   |
%e A244704   o
%p A244704 b:= proc(n, i, h, v) option remember; `if`(n=0,
%p A244704       `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
%p A244704       `if`(n=v, 1, add(binomial(A(i, min(i-1, h))+j-1, j)
%p A244704        *b(n-i*j, i-1, h, v-j), j=0..min(n/i, v)))))
%p A244704     end:
%p A244704 A:= proc(n, k) option remember;
%p A244704       `if`(n<2, n, add(b(n-1$2, j$2), j=1..min(k,n-1)))
%p A244704     end:
%p A244704 a:= n-> b(n-1$2, 3$2):
%p A244704 seq(a(n), n=4..50);
%t A244704 b[n_, i_, h_, v_] := b[n, i, h, v] = If[n == 0, If[v == 0, 1, 0], If[i < 1 || v < 1 || n < v, 0, If[n == v, 1, Sum[Binomial[A[i, Min[i - 1, h]] + j - 1, j]*b[n - i*j, i - 1, h, v - j], {j, 0, Min[n/i, v]}]]]];
%t A244704 A[n_, k_] := A[n, k] = If[n < 2, n, Sum[b[n - 1, n - 1, j, j], {j, 1, Min[k, n - 1]}]];
%t A244704 a[n_] := b[n-1, n-1, 3, 3];
%t A244704 a /@ Range[4, 50] (* _Jean-François Alcover_, Dec 27 2020, after _Alois P. Heinz_ *)
%Y A244704 Column k=3 of A244657.
%K A244704 nonn
%O A244704 4,3
%A A244704 _Alois P. Heinz_, Jul 04 2014