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

0,3

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

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.

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

Example

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 + ...

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 *)

Prog

(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 */

Crossrefs

Cf. A001190.

Sequence in context: A000968 A005908 A206119 * A317097 A252354 A052806

Adjacent sequences: A085745 A085746 A085747 * A085749 A085750 A085751

Keyword

nonn

Author

N. J. A. Sloane, Nov 07 2003

Status

approved