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A343937 Number of unlabeled semi-identity plane trees with n nodes.

%I A343937 #10 May 09 2021 06:12:07
%S A343937 1,1,2,5,13,38,117,375,1224,4095,13925,48006,167259,588189,2084948,
%T A343937 7442125,26725125,96485782,350002509,1275061385,4662936808,
%U A343937 17111964241,62996437297,232589316700,861028450579,3195272504259,11884475937910,44295733523881,165420418500155
%N A343937 Number of unlabeled semi-identity plane trees with n nodes.
%C A343937 In a semi-identity tree, only the non-leaf branches of any given vertex are required to be distinct. Alternatively, a rooted tree is a semi-identity tree iff the non-leaf branches of the root are all distinct and are themselves semi-identity trees.
%H A343937 Andrew Howroyd, <a href="/A343937/b343937.txt">Table of n, a(n) for n = 1..200</a>
%F A343937 G.f.: A(x) satisfies A(x) = x*Sum_{j>=0} j!*[y^j] exp(x*y - Sum_{k>=1} (-y)^k*(A(x^k) - x^k)/k). - _Andrew Howroyd_, May 08 2021
%e A343937 The a(1) = 1 through a(5) = 13 trees are the following. The number of nodes is the number of o's plus the number of brackets (...).
%e A343937   o  (o)  (oo)   (ooo)    (oooo)
%e A343937           ((o))  ((o)o)   ((o)oo)
%e A343937                  ((oo))   ((oo)o)
%e A343937                  (o(o))   ((ooo))
%e A343937                  (((o)))  (o(o)o)
%e A343937                           (o(oo))
%e A343937                           (oo(o))
%e A343937                           (((o))o)
%e A343937                           (((o)o))
%e A343937                           (((oo)))
%e A343937                           ((o(o)))
%e A343937                           (o((o)))
%e A343937                           ((((o))))
%t A343937 arsiq[n_]:=Join@@Table[Select[Union[Tuples[arsiq/@ptn]],#=={}||(UnsameQ@@DeleteCases[#,{}])&],{ptn,Join@@Permutations/@IntegerPartitions[n-1]}];
%t A343937 Table[Length[arsiq[n]],{n,10}]
%o A343937 (PARI)
%o A343937 F(p)={my(n=serprec(p,x)-1, q=exp(x*y + O(x*x^n))*prod(k=2, n, (1 + y*x^k + O(x*x^n))^polcoef(p,k,x)) ); sum(k=0, n, k!*polcoef(q,k,y))}
%o A343937 seq(n)={my(p=O(x)); for(n=1, n, p=x*F(p)); Vec(p)} \\ _Andrew Howroyd_, May 08 2021
%Y A343937 The not necessarily semi-identity version is A000108.
%Y A343937 The non-plane binary version is A063895, ranked by A339193.
%Y A343937 The non-plane version is A306200, ranked by A306202.
%Y A343937 The binary case is A343663.
%Y A343937 A000081 counts unlabeled rooted trees with n nodes.
%Y A343937 A001190*2 - 1 counts binary trees, ranked by A111299.
%Y A343937 A001190 counts semi-binary trees, ranked by A292050.
%Y A343937 A004111 counts identity trees, ranked by A276625.
%Y A343937 A306201 counts balanced semi-identity trees, ranked by A306203.
%Y A343937 A331966 counts lone-child avoiding semi-identity trees, ranked by A331965.
%Y A343937 Cf. A001678, A320230, A331934, A331963, A331964.
%K A343937 nonn
%O A343937 1,3
%A A343937 _Gus Wiseman_, May 07 2021
%E A343937 Terms a(17) and beyond from _Andrew Howroyd_, May 08 2021