A131764 Inverse Euler transform of central binomial coefficients A000984.

1, 2, 3, 10, 30, 102, 335, 1170, 4080, 14560, 52377, 190650, 698870, 2581110, 9586395, 35791358, 134215680, 505290270, 1908866960, 7233629130, 27487764474, 104715392730, 399822314775, 1529755308210, 5864061663920, 22517998136832, 86607683851185, 333599972392960, 1286742745883790, 4969489243995030, 19215358392200893, 74382032555280450, 288230376084602880

Offset: 0

Author

F. Chapoton, Oct 04 2007

Keywords

• nonn

Comments

This is the sequence of dimensions of a free Lie algebra on some specific set of generators.

Examples

			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 + ... .
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A022553, A000984, A000108.

Programs

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 *)

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 */

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

Extensions

More explicit definition from Michael Somos, Apr 01 2012. - N. J. A. Sloane, Feb 20 2017