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URL: https://oeis.org/A118206

⇱ A118206 - OEIS

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A118206
Euler transform of the Liouville function.
7
1, 1, 0, -1, 0, 0, 0, -1, -1, 0, 2, 0, -2, -2, 1, 2, 2, -2, -2, 0, 2, -1, -1, -2, 2, 5, 4, -5, -5, -2, 4, 2, -2, -7, 3, 8, 5, -7, -6, 1, 14, 4, -9, -14, 2, 5, 5, -10, -7, 6, 22, 3, -12, -20, 1, 15, 15, -16, -12, 4, 25, 6, -14, -31, 13, 33, 14, -39, -32, -6, 39, 15, -20, -31, 33, 41, 14, -53, -44, 3, 66, 12, -35, -51, 22, 48, 36, -60, -43, 21
OFFSET
0,11
FORMULA
G.f.: A(x) = Product_{k>=1} (1 - x^k)^(-lambda(k)) where lambda(k) is the Liouville function, A008836.
Logarithmic derivative yields A061020. - Paul D. Hanna, Sep 22 2011
G.f.: A(x) = Product_{k >= 1} C(2*k,x^k), where C(k,x) denotes the k-th cyclotomic polynomial. - Peter Bala, Mar 31 2023
MATHEMATICA
nmax = 100; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; CoefficientList[ Series[ Product[ (1 - x^k)^(-lambda[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ]
max = 100; s = Product[(1 - x^k)^(-LiouvilleLambda[k]), {k, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 23 2015 *)
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n, sumdiv(m, d, d*moebius(core(d)))*x^m/m)+x*O(x^n)), n)} /* Cf. A061020 - Paul D. Hanna, Sep 22 2011 */
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Stuart Clary, Apr 15 2006
STATUS
approved