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%I #19 Jan 26 2018 12:00:37
%S 1,1,2,3,6,10,21,38,78,153,314,632,1313,2700,5646,11786,24831,52348,
%T 111027,235834,502986,1074739,2303146,4944507,10639201,22930493,
%U 49511948,107065966,231874164,502834328,1091842824,2373565195,5165713137,11254029616,24542260010
%N Number of rooted trees on n nodes with thinning limbs.
%C A rooted tree with thinning limbs is such that if a node has k children, all its children have at most k children.
%H Alois P. Heinz, <a href="/A124343/b124343.txt">Table of n, a(n) for n = 1..141</a>
%e The a(5) = 6 trees are ((((o)))), (o((o))), (o(oo)), ((o)(o)), (oo(o)), (oooo). - _Gus Wiseman_, Jan 25 2018
%p b:= proc(n, i, h, v) option remember; `if`(n=0,
%p `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
%p `if`(n=v, 1, add(binomial(A(i, min(i-1, h))+j-1, j)
%p *b(n-i*j, i-1, h, v-j), j=0..min(n/i, v)))))
%p end:
%p A:= proc(n, k) option remember;
%p `if`(n<2, n, add(b(n-1$2, j$2), j=1..min(k,n-1)))
%p end:
%p a:= n-> A(n$2):
%p seq(a(n), n=1..35); # _Alois P. Heinz_, Jul 08 2014
%t 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 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 a[n_] := A[n, n];
%t Table[a[n], {n, 1, 35}] (* _Jean-François Alcover_, Mar 01 2016, after _Alois P. Heinz_ *)
%Y Cf. A000081, A032305, A124344-A124348, A290689, A298303, A298304, A298305, A298422.
%Y Row sums of A244657.
%K nonn
%O 1,3
%A _Christian G. Bower_, Oct 30 2006, suggested by _Franklin T. Adams-Watters_
%E More terms from _Alois P. Heinz_, Jul 04 2014
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