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A123864 Expansion of (eta(q^3) * eta(q^5))^2 / (eta(q) * eta(q^15)) in powers of q. 4
1, 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, 5, 1, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 6, 2, 4, 2, 6, 0, 2, 0, 3, 0, 4, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number 31 of the 74 eta-quotients listed in Table I of Martin (1996).

Multiplicative because this sequence is the inverse Moebius transform of a multiplicative sequence Kronecker(-15, n). - Andrew Howroyd, Jul 27 2018

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1000

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 [ 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -2, ...].

Moebius transform is period 15 sequence [ 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 0, ...].

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

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

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)).

a(15*n + 7) = a(15*n + 11) = a(15*n + 13) = a(15*n + 14) = 0. a(3*n) = a(n).

a(n) = A035175(n) unless n=0. a(n) = |A106406(n)| unless n=0.

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, Feb 10 2015

a(n) = Sum_{d | n}  Kronecker(-15, d). - Andrew Howroyd, Jul 27 2018

EXAMPLE

G.f. = 1 + 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[ (QPochhammer[ q^3] QPochhammer[ q^5])^2 / ( QPochhammer[ q] QPochhammer[ q^15]), {q, 0, n}]; (* Michael Somos, Feb 10 2015 *)

a[ n_] := If[ n < 1, Boole[n == 0], Sum[ KroneckerSymbol[ -15, d], { d, Divisors[ n]}]]; (* Michael Somos, Feb 10 2015 *)

PROG

(PARI) {a(n) = if( n<1, n==0, sumdiv( n, d, kronecker( -15, d)))};

(PARI) {a(n) = if( n<1, n==0, (qfrep( [2, 1; 1, 8], n, 1) + qfrep( [4, 1; 1, 4], n, 1))[n])};

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

(MAGMA) A := Basis( ModularForms( Gamma1(15), 1), 106); A[1] + A[2] + 2*A[3] + A[4] + 3*A[5] + A[6] + 2*A[7]; /* Michael Somos, Feb 10 2015 */

CROSSREFS

Cf. A035175, A106406.

Sequence in context: A156248 A324817 A106406 * A035175 A092412 A265578

Adjacent sequences:  A123861 A123862 A123863 * A123865 A123866 A123867

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Oct 14 2006

STATUS

approved

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Last modified March 8 19:37 EST 2021. Contains 341953 sequences. (Running on oeis4.)