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.

1, 1, 2, 4, 9, 20, 46, 106, 248, 582, 1376, 3264, 7777, 18581, 44526, 106936, 257379, 620577, 1498788, 3625026, 8779271, 21287278, 51671864, 125550018, 305333281, 743179460, 1810290446, 4412783988, 10763786019, 26271534125, 64158771500, 156769178340

Offset: 0

Author

N. J. A. Sloane, Nov 07 2003

Keywords

• nonn

Comments

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

Examples

			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 + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2536
• M. R. Bremner, L. A. Peresi and J. Sanchez-Ortega, Malcev dialgebras, arXiv preprint arXiv:1108.0586 [math.RA], 2011.
• Chloe E. Shiff and Noah A. Rosenberg, Enumeration of rooted binary perfect phylogenies, arXiv:2410.14915 [q-bio.PE], 2024. See pp. 9, 17.

Crossrefs

Cf. A001190.

Programs

Maple

b:= proc(n) option remember; `if`(n<2, n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(b(n/2)))+add(b(i)*b(n-i), i=1..n/2))
    end:
a:= proc(n) option remember; `if`(n<1, 1,
      add(a(n-i)*b(i), i=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 07 2017

Mathematica

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}] ];
a[n_] := a[n] = If[n < 1, 1, Sum[a[n - i]*b[i], {i, 1, n}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *)

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 */

Formula

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