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A292087 Limit of the number of (unlabeled) rooted trees without unary nodes where n is the difference between the number of leafs and the maximal outdegree as the tree size increases.

%I A292087 #15 Feb 28 2024 08:45:46
%S A292087 1,2,7,23,78,262,893,3040,10411,35724,122950,424004,1465254,5071981,
%T A292087 17584226,61046464,212197118,738422362,2572261241,8968726829,
%U A292087 31298189180,109307655964,382031357974,1336107044159,4675807680776,16372936282017,57363325974309
%N A292087 Limit of the number of (unlabeled) rooted trees without unary nodes where n is the difference between the number of leafs and the maximal outdegree as the tree size increases.
%H A292087 Alois P. Heinz, <a href="/A292087/b292087.txt">Table of n, a(n) for n = 0..100</a>
%H A292087 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%e A292087 : a(0) = 1:
%e A292087 :                  o
%e A292087 :               //( )\\
%e A292087 :             N N N N N N
%e A292087 :
%e A292087 : a(1) = 2:
%e A292087 :             o               o
%e A292087 :           /   \          / /|\ \
%e A292087 :          o     N        o N N N N
%e A292087 :       / /|\ \          ( )
%e A292087 :      N N N N N         N N
%e A292087 :
%e A292087 : a(2) = 7:
%e A292087 :           o             o            o             o
%e A292087 :          / \          /   \        /( )\         / | \
%e A292087 :         o   N        o     N      o N N N       o  N  N
%e A292087 :        / \         /( )\         / \          /( )\
%e A292087 :       o   N       o N N N       o   N        N N N N
%e A292087 :     /( )\        ( )           ( )
%e A292087 :    N N N N       N N           N N
%e A292087 :
%e A292087 :          o            o              o
%e A292087 :        /   \        /( )\         / ( \ \
%e A292087 :       o     o      o N N N      o   o  N N
%e A292087 :     /( )\  ( )    /|\          ( ) ( )
%e A292087 :    N N N N N N   N N N         N N N N
%e A292087 :
%p A292087 b:= proc(n, i, v, k) option remember; `if`(n=0,
%p A292087       `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
%p A292087       `if`(v=n, 1, add(binomial(A(i,k)+j-1, j)*
%p A292087        b(n-i*j, i-1, v-j, k), j=0..min(n/i, v)))))
%p A292087     end:
%p A292087 A:= proc(n, k) option remember; `if`(n<2, n,
%p A292087       add(b(n, n+1-j, j, k), j=2..min(n, k)))
%p A292087     end:
%p A292087 a:= n-> A(2*n+3, n+3)-A(2*n+3, n+2):
%p A292087 seq(a(n), n=0..23);
%t A292087 b[n_, i_, v_, k_] := b[n, i, v, k] = If[n == 0,
%t A292087     If[v == 0, 1, 0], If[i < 1 || v < 1 || n < v, 0,
%t A292087     If[v == n, 1, Sum[Binomial[A[i, k] + j - 1, j]*
%t A292087     b[n - i*j, i - 1, v - j, k], {j, 0, Min[n/i, v]}]]]];
%t A292087 A[n_, k_] := A[n, k] = If[n < 2, n,
%t A292087     Sum[b[n, n + 1 - j, j, k], {j, 2, Min[n, k]}]];
%t A292087 a[n_] := A[2*n + 3, n + 3] - A[2*n + 3, n + 2];
%t A292087 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Feb 28 2024, after _Alois P. Heinz_ *)
%Y A292087 Limit of reversed rows of A292086.
%K A292087 nonn
%O A292087 0,2
%A A292087 _Alois P. Heinz_, Sep 08 2017