A159818 Expansion of f(q) * f(q^5) in powers of q where f() is a Ramanujan theta function.

1, 1, -1, 0, 0, 0, 1, -2, 0, 0, -2, -1, -1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0, 0, -2, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 1, 0, 0, -2, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -1, -2, 0, 0, -2, -1, 0, 0, 0, 2, 0, 2, 0, 0, -2, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 1, -2, 0, 0

Offset

0,8

Comments

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

Number 71 of the 74 eta-quotients listed in Table I of Martin (1996).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/4) * eta(q^2)^3 * eta(q^10)^3 / (eta(q) * eta(q^4) * eta(q^5) * eta(q^20)) in powers of q.

Euler transform of period 20 sequence [ 1, -2, 1, -1, 2, -2, 1, -1, 1, -4, 1, -1, 1, -2, 2, -1, 1, -2, 1, -2, ...].

a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(5^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p == 11, 13, 17, 19 (mod 20), b(p^e) = (-1)^(e/2) * (1 + (-1)^e) / 2 if p == 3, 7 (mod 20), b(p^e) = (-1)^(e*z) * (e+1) if p == 1, 9 (mod 20) where p = x^2 + 5*y^2 and z = 1 if x or y == 0 (mod 4) else z = 0.

G.f. is a period 1 Fourier series which satisfies f(-1 / (320 t)) = (320)^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

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

a(n) = (-1)^n * A030202(n). Convolution square is A159817. a(5*n + 3) = a(5*n + 4) = a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = -a(n).

Example

G.f. = 1 + x - x^2 + x^6 - 2*x^7 - 2*x^10 - x^11 - x^12 + 2*x^15 + x^20 + ...
G.f. = q + q^5 - q^9 + q^25 - 2*q^29 - 2*q^41 - q^45 - q^49 + 2*q^61 + q^81 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ -x^5], {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)

Prog

(PARI) {a(n) = my(A, p, e, x, z); if(n<0, 0, n = 4*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 0, p==5, 1, p%20>10, !(e%2), p%4==3, kronecker(-4, e+1), for(y=1, sqrtint(p\5), if(issquare(p - 5*y^2, &x), z = if(x%2, y, x)%4/2; break)); (-1)^(e*z) *(e+1))))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^10 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^20 + A)), n))};

Crossrefs

Cf. A030202, A159817.

Sequence in context: A156996 A029304 A030202 * A081827 A100286 A280912

Adjacent sequences: A159815 A159816 A159817 * A159819 A159820 A159821

Keyword

sign

Author

Michael Somos, Apr 22 2009

Status

approved