OFFSET
0,2
COMMENTS
This is the sequence of dimensions of a free Lie algebra on some specific set of generators.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
FORMULA
a(n) = (1/n) * Sum_{d|n} moebius(n/d)*2^(2*d-1) for n > 0, a(0) = 1.
a(n) ~ 2^(2*n-1) / n. - Vaclav Kotesovec, Oct 09 2019
EXAMPLE
2*x + 3*x^2 + 10*x^3 + 30*x^4 + 102*x^5 + 335*x^6 + 1170*x^7 + 4080*x^8 + ...
(1-x)^(-2)*(1-x^2)^(-3)*(1-x^3)^(-10)*(1-x^4)^(-30)*(1-x^5)^(-102) = 1 + 2*x + 6*x^2 + 20*x^3 + 70*x^4 + 252*x^5 + ... .
MAPLE
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> binomial(2*n, n)):
seq(a(n), n = 0..32); # Peter Luschny, Nov 21 2022
MATHEMATICA
a[n_] := (1/n)*DivisorSum[n, MoebiusMu[n/#]*2^(2*#-1)&]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Feb 20 2017 *)
PROG
(MuPAD) a(n):=proc(n) begin 1/n*_plus(moebius(n/d)*2^(2*d-1)$d in divisors(n)) end;
(PARI) a(n)=sumdiv(n, d, 1/n*moebius(n/d)*2^(d*2-1)); /* Joerg Arndt, Jul 06 2011 */
(PARI) {a(n) = local(A); if( n<1, 0, A = sqrt(1 - 4*x + x * O(x^n)); for( k=1, n-1, A *= (1 - x^k + x * O(x^n))^ polcoeff( A, k)); -polcoeff( A, n))} /* Michael Somos, Apr 01 2012 */
CROSSREFS
KEYWORD
nonn
AUTHOR
F. Chapoton, Oct 04 2007
EXTENSIONS
More explicit definition from Michael Somos, Apr 01 2012. - N. J. A. Sloane, Feb 20 2017
STATUS
approved