A106406 - OEIS

%I #31 Sep 08 2022 08:45:18

%S 1,-2,-1,3,-1,2,0,-4,1,2,0,-3,0,0,1,5,-2,-2,2,-3,0,0,-2,4,1,0,-1,0,0,

%T -2,2,-6,0,4,0,3,0,-4,0,4,0,0,0,0,-1,4,-2,-5,1,-2,2,0,-2,2,0,0,-2,0,0,

%U 3,2,-4,0,7,0,0,0,-6,2,0,0,-4,0,0,-1,6,0,0,2

%N Expansion of (eta(q) * eta(q^15))^2 / (eta(q^3) * eta(q^5)) in powers of q.

%C Number 30 of the 74 eta-quotients listed in Table I of Martin (1996).

%H G. C. Greubel, <a href="/A106406/b106406.txt">Table of n, a(n) for n = 1..10000</a>

%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>

%F Euler transform of period 15 sequence [-2, -2, -1, -2, -1, -1, -2, -2, -1, -1, -2, -1, -2, -2, -2, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -v^3 + 4 * u*v*w + 2 * u*w^2 + u^2*w.

%F a(n) is multiplicative with a(3^e) = a(5^e) = (-1)^e, a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11, 13, 14 (mod 15), a(p^e) = e+1 if p == 1, 4 (mod 15), a(p^e) = (e+1) * (-1)^e if p == 2, 8 (mod 15). - _Michael Somos_, Oct 19 2005

%F G.f.: (1/2) * (Sum_{n,m in Z} x^(n^2 + n*m + 4*m^2) - x^(2*n^2 + n*m + 2 *m^2)). - _Michael Somos_, Aug 25 2006

%F G.f.: Sum_{k>0} Kronecker(k, 3) * x^k * (1 - x^k) * (1 - x^(2*k)) / (1 - x^(5*k)) = Sum_{k>0} Kronecker(k, 5) * x^k * (1 - x^k) / (1 - x^(3*k)).

%F G.f.: x * Product_{k>0} ((1 - x^k) * (1 - x^(15*k)))^2 / ((1 - x^(3*k)) * (1 - x^(5*k))).

%F a(15*n + 7) = a(15*n + 11) = a(15*n + 13) = a(15*n + 14) = 0. a(3*n) = a(5*n) = -a(n).

%F A035175(n) = |a(n)|. a(n)>0 iff n in A028957. a(n)<0 iff n in A028955.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = 15^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - _Michael Somos_, May 18 2015

%e G.f. = q - 2*q^2 - q^3 + 3*q^4 - q^5 + 2*q^6 - 4*q^8 + q^9 + 2*q^10 + ...

%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^15])^2 / (QPochhammer[ q^3] QPochhammer[ q^5]), {q, 0, n}]; (* _Michael Somos_, May 18 2015 *)

%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ #, 3] KroneckerSymbol[ n/#, 5] &]]; (* _Michael Somos_, May 18 2015 *)

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^15 + A)^2 / (eta(x^3 + A) * eta(x^5 + A)), n))};

%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( d, 3) * kronecker( n/d, 5)))};

%o (PARI) {a(n) = my(A, p, e, x); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==3 || p==5, (-1)^e, (p%15) != 2^(x = valuation( p%15, 2)), (e+1)%2, (e+1) * (-1)^(x*e))))};

%o (PARI) {a(n) = if( n<1, 0, (qfrep([2, 1;1, 8],n, 1) - qfrep([4, 1;1, 4], n, 1))[n])}; /* _Michael Somos_, Aug 25 2006 */

%o (Magma) A := Basis( ModularForms( Gamma1(15), 1), 80); A[2] - 2*A[3] - A[4] + 3*A[5] - A[6] + 2*A[7]; /* _Michael Somos_, May 18 2015 */

%Y Cf. A028955, A028957, A035175, A123864.

%K sign,mult

%O 1,2

%A _Michael Somos_, May 02 2005