A229502 Expansion of q * f(-q) * f(-q^4) * f(-q^16) * f(q, -q^3) in powers of q where f() is a Ramanujan theta function.

1, 0, -2, -2, 0, 2, 2, 4, -1, -2, 0, 0, -2, -4, 2, -4, -2, 2, 6, 4, 0, 2, -6, -4, 3, 2, 0, 4, 6, 0, -8, 0, -2, 0, -4, -2, 0, -6, 2, -4, 0, 4, 0, -4, 2, 12, 8, 8, 3, -6, 4, 0, 0, -8, 2, 0, -6, -6, 0, -4, -18, 0, 2, 8, 2, 0, -10, 4, 0, 4, 10, 0, 4, 6, 0, 0, 4

Offset

1,3

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..2500 (terms 1..77 from Michael Somos)

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Euler transform of period 16 sequence [0, -2, -2, -1, -2, -1, 0, -4, 0, -1, -2, -1, -2, -2, 0, -4, ...].

a(2^n) = A090132(n). a(16*n + 5) = a(16*n + 11) = 0. 2 * a(n) = A229893(8*n). a(2*n) = -2 * A229893(n).

Example

G.f. = q - 2*q^3 - 2*q^4 + 2*q^6 + 2*q^7 + 4*q^8 - q^9 - 2*q^10 - 2*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^3, -q^4] QPochhammer[ -q, -q^4] QPochhammer[ -q^4] QPochhammer[ q] QPochhammer[ q^4] QPochhammer[ q^16], {q, 0, n}];

Prog

(PARI) {a(n) = my(A, m); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^16 + A) * sum( k=0, n, if( issquare( 8*k + 1, &m), (-1)^((m\2 + 2) \ 4) * x^k, 0), A), n))};
(Sage) CuspForms( Gamma1(16), 2, prec=78).0;
(Magma) Basis( CuspForms( Gamma1(16), 2), 78)[1];

Crossrefs

Cf. A090132, A229893.

Sequence in context: A177225 A236306 A153239 * A356359 A141661 A278521

Adjacent sequences: A229499 A229500 A229501 * A229503 A229504 A229505

Keyword

sign

Author

Michael Somos, Oct 02 2013

Status

approved