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A131764 Inverse Euler transform of central binomial coefficients A000984. 1
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 (list; graph; refs; listen; history; text; internal format)
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 + ... .

MATHEMATICA

a[n_] := (1/n)*DivisorSum[n, MoebiusMu[n/#]*2^(2*#-1)&]; Table[a[n], {n, 1, 32}] (* Jean-Fran├žois 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 A209003 A080022

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

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Last modified June 19 15:03 EDT 2021. Contains 345141 sequences. (Running on oeis4.)