A129448 - OEIS

A129448

Expansion of q * psi(-q) * chi(q^3)^2 * psi(-q^9) in powers of q where psi(), chi() are Ramanujan theta functions.

1, -1, 0, 1, -2, 0, 0, -1, 0, 2, 0, 0, 2, 0, 0, 1, -2, 0, 0, -2, 0, 0, 0, 0, 3, -2, 0, 0, -2, 0, 0, -1, 0, 2, 0, 0, 2, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 1, -3, 0, 2, -2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, -4, 0, 0, -2, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, -2, 0, 2, 0, 0, 4, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 2, -1, 0, 3, -2, 0, 0, -2, 0

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OFFSET

1,5

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Number 50 of the 74 eta-quotients listed in Table I of Martin (1996).

LINKS

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of eta(q) * eta(q^4) * eta(q^6)^4 * eta(q^9) * eta(q^36) / (eta(q^2) * eta(q^3)^2 * eta(q^12)^2 * eta(q^18)) in powers of q.

Euler transform of period 36 sequence [ -1, 0, 1, -1, -1, -2, -1, -1, 0, 0, -1, -1, -1, 0, 1, -1, -1, -2, -1, -1, 1, 0, -1, -1, -1, 0, 0, -1, -1, -2, -1, -1, 1, 0, -1, -2, ...].

a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = 0^e, a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12), a(p^e) = e+1 if p == 1 (mod 12), a(p^e) = (-1)^e * (e+1) if p == 5 (mod 12).

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) f(t) where q = exp(2 Pi i t).

G.f.: Sum_{k>0} Kronecker(12, k) * x^k/ (1 + x^k + x^(2*k)).

|a(n)| = A091400(n). a(3*n) = a(4*n + 3) = 0. a(2*n) = -a(n). a(3*n + 1) = A122865(n). a(3*n + 2) = - A122856(n). a(4*n + 1) = A121363(n). a(12*n + 1) = A002175(n). a(12*n + 5) = -2 * A121444(n).

EXAMPLE

G.f. = q - q^2 + q^4 - 2*q^5 - q^8 + 2*q^10 + 2*q^13 + q^16 - 2*q^17 - 2*q^20 + ...

MATHEMATICA

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ 12, d] KroneckerSymbol[ -3, n/d], {d, Divisors[ n]}]]; (* Michael Somos, Jul 09 2015 *)

a[ n_] := SeriesCoefficient[ q QPochhammer[ -q^3, q^6]^2 EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(9/2)] / (2 q^(5/4)), {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)

PROG

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker( 12, d) * kronecker( -3, n/d)))};

(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, p==2, (-1)^e, p%12>6, !(e%2), (-1)^(e * (p%12==5)) * (e+1))))};

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^4 * eta(x^9 + A) * eta(x^36 + A) / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2 * eta(x^18 + A)), n))};

(Magma) A := Basis( ModularForms( Gamma1(36), 1), 82); A[2] - A[3] + A[5] - 2*A[6] - A[9] + 2*A[11] + 2*A[14] + A[17] - 2*A[18]; /* Michael Somos, Jul 09 2015 */

CROSSREFS

Cf. A002175, A091400, A121363, A121444, A122856, A122865.

Sequence in context: A063725 A084888 A091400 * A239003 A123759 A072453

Adjacent sequences: A129445 A129446 A129447 * A129449 A129450 A129451

KEYWORD

sign,mult

AUTHOR

Michael Somos, Apr 16 2007

STATUS

approved