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 A097109 G.f.: s(2)^2*s(3)^3/(s(1)*s(6)^2), where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. 4
 1, 1, 0, -2, -3, 0, 0, 2, 0, -2, 0, 0, 6, 2, 0, 0, -3, 0, 0, 2, 0, -4, 0, 0, 0, 1, 0, -2, -6, 0, 0, 2, 0, 0, 0, 0, 6, 2, 0, -4, 0, 0, 0, 2, 0, 0, 0, 0, 6, 3, 0, 0, -6, 0, 0, 0, 0, -4, 0, 0, 0, 2, 0, -4, -3, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, -2, -6, 0, 0, 2, 0, -2, 0, 0, 12, 0, 0, 0, 0, 0, 0, 4, 0, -4, 0, 0, 0, 2, 0, 0, -3, 0, 0, 2, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Coefficients are multiplicative [Fine]. REFERENCES N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 80, Eq. (32.36). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 FORMULA Fine gives an explicit formula for a(n) in terms of the divisors of n. Expansion of (a(q) - 3*a(q^3) - 4*a(q^4) + 12*a(q^12)) / 6 in powers of q where a() is a cubic AGM theta function. - Michael Somos, Sep 15 2006 Euler transform of period 6 sequence [ 1, -1, -2, -1, 1, -2, ...]. - Michael Somos, Sep 15 2006 a(n) is multiplicative with a(2^e) = -3(1+(-1)^e)/2 if e>0, a(3^e) = -2 if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6). - Michael Somos, Sep 15 2006 a(3*n + 2) = 0. a(3*n) = A115978(n). a(3*n + 1) = A122861(n). MATHEMATICA QP = QPochhammer; s = QP[q^2]^2*(QP[q^3]^3/(QP[q]*QP[q^6]^2)) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015, adapted from PARI *) PROG (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^2), n))} /* Michael Somos, Sep 15 2006 */ (PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if(p==2, 3*(e%2-1), if( p==3, -2, if( p%6==1, e+1, 1-e%2))))))} /* Michael Somos, Sep 15 2006 */ CROSSREFS Cf. A115978, A122861. Sequence in context: A167613 A319753 A078332 * A160018 A175099 A151867 Adjacent sequences: A097106 A097107 A097108 * A097110 A097111 A097112 KEYWORD sign,mult AUTHOR N. J. A. Sloane, Sep 16 2004 STATUS approved

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Last modified September 22 10:41 EDT 2023. Contains 365520 sequences. (Running on oeis4.)