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A245121 Number of n-node rooted identity trees with thinning limbs and root outdegree (branching factor) 2.

%I A245121 #17 Dec 28 2020 09:52:22
%S A245121 1,1,3,4,8,12,22,36,63,107,188,327,578,1020,1820,3248,5839,10511,
%T A245121 19022,34484,62755,114421,209234,383327,703901,1294822,2386376,
%U A245121 4405083,8144701,15080416,27961728,51912054,96496481,179577543,334558479,623936240,1164765120
%N A245121 Number of n-node rooted identity trees with thinning limbs and root outdegree (branching factor) 2.
%C A245121 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 A245121 Alois P. Heinz, <a href="/A245121/b245121.txt">Table of n, a(n) for n = 4..1000</a>
%F A245121 a(n) ~ c * d^n / n^(3/2), where d = 1.938950593419038561279875... and c = 0.929315638487153276953929... . - _Vaclav Kotesovec_, Jul 13 2014
%e A245121 a(7) = 4:
%e A245121 :   o     o       o      o   :
%e A245121 :  / \   / \     / \    / \  :
%e A245121 : o   o o   o   o   o  o   o :
%e A245121 : |     |   |  / \    ( )  | :
%e A245121 : o     o   o o   o   o o  o :
%e A245121 : |     |     |       |      :
%e A245121 : o     o     o       o      :
%e A245121 : |     |     |              :
%e A245121 : o     o     o              :
%e A245121 : |                          :
%e A245121 : o                          :
%p A245121 b:= proc(n, i, h, v) option remember; `if`(n=0, `if`(v=0, 1, 0),
%p A245121       `if`(i<1 or v<1 or n<v, 0, add(binomial(A(i, min(i-1, h)), j)
%p A245121        *b(n-i*j, i-1, h, v-j), j=0..min(n/i, v))))
%p A245121     end:
%p A245121 A:= proc(n, k) option remember;
%p A245121       `if`(n<2, n, add(b(n-1$2, j$2), j=1..min(k, n-1)))
%p A245121     end:
%p A245121 a:= n-> b(n-1$2, 2$2):
%p A245121 seq(a(n), n=4..45);
%t A245121 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, Sum[Binomial[A[i, Min[i - 1, h]], j] b[n - i*j, i - 1, h, v - j], {j, 0, Min[n/i, v]}]]];
%t A245121 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 A245121 a[n_] := b[n-1, n-1, 2, 2];
%t A245121 a /@ Range[4, 45] (* _Jean-François Alcover_, Dec 28 2020, after _Alois P. Heinz_ *)
%Y A245121 Column k=2 of A245120.
%K A245121 nonn
%O A245121 4,3
%A A245121 _Alois P. Heinz_, Jul 12 2014