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A085748 G.f.: 1/(1-G001190), where G001190 = x + x^2 + x^3 + 2x^4 + 3x^5 + ... is the g.f. for the Wedderburn-Etherington numbers A001190.

%I A085748 #37 Oct 28 2024 17:09:03
%S A085748 1,1,2,4,9,20,46,106,248,582,1376,3264,7777,18581,44526,106936,257379,
%T A085748 620577,1498788,3625026,8779271,21287278,51671864,125550018,305333281,
%U A085748 743179460,1810290446,4412783988,10763786019,26271534125,64158771500,156769178340
%N A085748 G.f.: 1/(1-G001190), where G001190 = x + x^2 + x^3 + 2x^4 + 3x^5 + ... is the g.f. for the Wedderburn-Etherington numbers A001190.
%C A085748 a(n) is also the number of interpretations of c*x^n (or number of ways to insert parentheses) when multiplication is commutative but not associative. E.g. a(2) = 2: c(x^2) and (c.x)x. a(3)=4: c(x.x^2), (c.x)(x^2), (c.x^2)x, and ((c.x)x)x. - _Paul Zimmermann_, Dec 04 2009
%H A085748 Alois P. Heinz, <a href="/A085748/b085748.txt">Table of n, a(n) for n = 0..2536</a>
%H A085748 M. R. Bremner, L. A. Peresi and J. Sanchez-Ortega, <a href="http://arxiv.org/abs/1108.0586">Malcev dialgebras</a>, arXiv preprint arXiv:1108.0586 [math.RA], 2011.
%H A085748 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. 9, 17.
%F A085748 G.f. A(x) satisfies: x * A(x)^2 = B(x * A(x^2)) where B(x) = x / (1 - 2*x). - _Michael Somos_, Feb 17 2004
%e A085748 G.f. = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 46*x^6 + 106*x^7 + 248*x^8 + ...
%p A085748 b:= proc(n) option remember; `if`(n<2, n, `if`(n::odd, 0,
%p A085748       (t-> t*(1-t)/2)(b(n/2)))+add(b(i)*b(n-i), i=1..n/2))
%p A085748     end:
%p A085748 a:= proc(n) option remember; `if`(n<1, 1,
%p A085748       add(a(n-i)*b(i), i=1..n))
%p A085748     end:
%p A085748 seq(a(n), n=0..25);  # _Alois P. Heinz_, Sep 07 2017
%t A085748 b[n_] := b[n] = If[n < 2, n, If[OddQ[n], 0, Function[t, t*(1 - t)/2][ b[n/2] ] ] + Sum[b[i]*b[n - i], {i, 1, n/2}] ];
%t A085748 a[n_] := a[n] = If[n < 1, 1, Sum[a[n - i]*b[i], {i, 1, n}]];
%t A085748 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Nov 06 2017, after _Alois P. Heinz_ *)
%o A085748 (PARI) {a(n) = local(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = sqrt( subst( x / (1 - 2*x), x, x * subst(A, x, x^2)) / x)); polcoeff(A, n))}; /* _Michael Somos_, Feb 17 2004 */
%Y A085748 Cf. A001190.
%K A085748 nonn
%O A085748 0,3
%A A085748 _N. J. A. Sloane_, Nov 07 2003