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%I #7 Oct 28 2024 17:09:06
%S 1,1,2,3,7,14,35,85,226,600,1658,4622,13141,37699,109419,320017,
%T 943329,2797788,8346030,25019401,75340824,227777899,691146578,
%U 2104028507,6424449318,19670277332,60378290912,185763773723,572764664975
%N Number of binary trees of weight n where leaves have positive integer weights, where the order of subtrees is insignificant. Commutative non-associative version of partitions of n.
%H Chloe E. Shiff and Noah A. Rosenberg, <a href="https://arxiv.org/abs/2410.14915">Enumeration of rooted binary perfect phylogenies</a>, arXiv:2410.14915 [q-bio.PE], 2024. See pp. 5, 9, 17.
%F a(2n) = 1 + C(a(n)+1, 2) + sum_{k=1}^{n/2-1} a(k)*a(2n-k). a(2n+1) = 1 + sum_{k=1}^{(n-1)/2} a(k)*a(2n+1-k), with a(0)=0.
%e For a(4)=7, we have the following 7 sums: 4, 3+1, 2+2, (2+1)+1, (1+1)+2, ((1+1)+1)+1, (1+1)+(1+1).
%Y Cf. A007317, A000041.
%K easy,nonn
%O 0,3
%A _Franklin T. Adams-Watters_, Jan 23 2006