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A244712 Number of 2n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) n.

%I A244712 #12 Feb 26 2017 09:59:24
%S A244712 0,1,1,3,6,15,31,74,159,365,805,1819,4041,9091,20274,45474,101644,
%T A244712 227755,509559,1141446,2555232,5723626,12817678,28713594,64319189,
%U A244712 144104857,322867573,723482538,1621264326,3633487621,8143682973,18253865341,40918282628,91730206467
%N A244712 Number of 2n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) n.
%C A244712 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 A244712 Vaclav Kotesovec, <a href="/A244712/b244712.txt">Table of n, a(n) for n = 0..193</a> (terms 0..70 from Alois P. Heinz)
%F A244712 a(n) = A244657(2n,n).
%p A244712 b:= proc(n, i, h, v) option remember; `if`(n=0,
%p A244712       `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
%p A244712       `if`(n=v, 1, add(binomial(A(i, min(i-1, h))+j-1, j)
%p A244712        *b(n-i*j, i-1, h, v-j), j=0..min(n/i, v)))))
%p A244712     end:
%p A244712 A:= proc(n, k) option remember;
%p A244712       `if`(n<2, n, add(b(n-1$2, j$2), j=1..min(k,n-1)))
%p A244712     end:
%p A244712 a:= n-> b(2*n-1$2, n$2):
%p A244712 seq(a(n), n=0..40);
%t A244712 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 A244712 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 A244712 a[n_] := b[2n-1, 2n-1, n, n]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Feb 26 2017, after _Alois P. Heinz_ *)
%Y A244712 Cf. A244657.
%K A244712 nonn
%O A244712 0,4
%A A244712 _Alois P. Heinz_, Jul 04 2014