cp's OEIS Frontend

A244703 Number of n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) 2.

%I A244703 #15 Dec 27 2020 19:44:12
%S A244703 1,1,3,4,9,13,26,42,81,138,262,467,885,1620,3076,5743,10953,20721,
%T A244703 39714,75873,146139,281259,544230,1053552,2047147,3981790,7766018,
%U A244703 15165195,29676887,58148087,114129308,224278526,441368913,869583189,1715365690,3387344619
%N A244703 Number of n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) 2.
%C A244703 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 A244703 Alois P. Heinz, <a href="/A244703/b244703.txt">Table of n, a(n) for n = 3..1000</a>
%F A244703 a(n) ~ c * d^n / n^(3/2), where d = 2.0554620926822709065075792..., c = 1.0209036918758320315742... . - _Vaclav Kotesovec_, Aug 27 2014
%e A244703 a(6) = 4:
%e A244703     o        o      o        o
%e A244703    / \      / \    / \      / \
%e A244703   o   o    o   o  o   o    o   o
%e A244703   |       / \     |   |   / \  |
%e A244703   o      o   o    o   o  o   o o
%e A244703   |      |        |
%e A244703   o      o        o
%e A244703   |
%e A244703   o
%p A244703 b:= proc(n, i, h, v) option remember; `if`(n=0,
%p A244703       `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
%p A244703       `if`(n=v, 1, add(binomial(A(i, min(i-1, h))+j-1, j)
%p A244703        *b(n-i*j, i-1, h, v-j), j=0..min(n/i, v)))))
%p A244703     end:
%p A244703 A:= proc(n, k) option remember;
%p A244703       `if`(n<2, n, add(b(n-1$2, j$2), j=1..min(k,n-1)))
%p A244703     end:
%p A244703 a:= n-> b(n-1$2, 2$2):
%p A244703 seq(a(n), n=3..50);
%t A244703 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 A244703 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 A244703 a[n_] := b[n-1, n-1, 2, 2];
%t A244703 a /@ Range[3, 50] (* _Jean-François Alcover_, Dec 27 2020, after _Alois P. Heinz_ *)
%Y A244703 Column k=2 of A244657.
%K A244703 nonn
%O A244703 3,3
%A A244703 _Alois P. Heinz_, Jul 04 2014