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
KEYWORD
nonn,changed
AUTHOR
N. J. A. Sloane, Nov 07 2003
STATUS
approved