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%I #51 Nov 21 2022 09:12:34
%S 1,2,3,10,30,102,335,1170,4080,14560,52377,190650,698870,2581110,
%T 9586395,35791358,134215680,505290270,1908866960,7233629130,
%U 27487764474,104715392730,399822314775,1529755308210,5864061663920,22517998136832,86607683851185,333599972392960,1286742745883790,4969489243995030,19215358392200893,74382032555280450,288230376084602880
%N Inverse Euler transform of central binomial coefficients A000984.
%C This is the sequence of dimensions of a free Lie algebra on some specific set of generators.
%H Seiichi Manyama, <a href="/A131764/b131764.txt">Table of n, a(n) for n = 0..1000</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F a(n) = (1/n) * Sum_{d|n} moebius(n/d)*2^(2*d-1) for n > 0, a(0) = 1.
%F a(n) ~ 2^(2*n-1) / n. - _Vaclav Kotesovec_, Oct 09 2019
%e 2*x + 3*x^2 + 10*x^3 + 30*x^4 + 102*x^5 + 335*x^6 + 1170*x^7 + 4080*x^8 + ...
%e (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 + ... .
%p # The function EulerInvTransform is defined in A358451.
%p a := EulerInvTransform(n -> binomial(2*n, n)):
%p seq(a(n), n = 0..32); # _Peter Luschny_, Nov 21 2022
%t a[n_] := (1/n)*DivisorSum[n, MoebiusMu[n/#]*2^(2*#-1)&]; Table[a[n], {n, 1, 32}] (* _Jean-François Alcover_, Feb 20 2017 *)
%o (MuPAD) a(n):=proc(n) begin 1/n*_plus(moebius(n/d)*2^(2*d-1)$d in divisors(n)) end;
%o (PARI) a(n)=sumdiv(n,d,1/n*moebius(n/d)*2^(d*2-1)); /* _Joerg Arndt_, Jul 06 2011 */
%o (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 */
%Y Cf. A022553, A000984, A000108.
%K nonn
%O 0,2
%A _F. Chapoton_, Oct 04 2007
%E More explicit definition from _Michael Somos_, Apr 01 2012. - _N. J. A. Sloane_, Feb 20 2017
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