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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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.

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-Fran├žois 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

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

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Last modified March 19 11:10 EDT 2019. Contains 321329 sequences. (Running on oeis4.)