A106406 Expansion of (eta(q) * eta(q^15))^2 / (eta(q^3) * eta(q^5)) in powers of q.

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

Offset

1,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Formula

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

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.

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

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

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)).

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

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).

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

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

Example

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

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^15])^2 / (QPochhammer[ q^3] QPochhammer[ q^5]), {q, 0, n}]; (* Michael Somos, May 18 2015 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ #, 3] KroneckerSymbol[ n/#, 5] &]]; (* Michael Somos, May 18 2015 *)

Prog

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

Crossrefs

Cf. A028955, A028957, A035175, A123864.

Sequence in context: A207031 A156248 A324817 * A123864 A035175 A092412

Adjacent sequences: A106403 A106404 A106405 * A106407 A106408 A106409

Keyword

sign,mult

Author

Michael Somos, May 02 2005

Status

approved