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

0,2

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

Cf. A022553, A000984, A000108.

Sequence in context: A319671 A338593 A270363 * A066706 A353052 A209003

Adjacent sequences: A131761 A131762 A131763 * A131765 A131766 A131767

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