A227618 Expansion of q * f(-q) * f(-q^4) * f(-q^16) * f(-q^6, -q^10) in powers of q where f() is a Ramanujan theta function.

1, -1, -1, 0, -1, 2, 0, 2, 0, 0, 1, -2, -1, -2, 2, -4, -2, 1, 3, 2, 2, 0, 0, 0, 0, 2, -4, 4, 3, -2, -8, 4, -2, 2, -2, -2, 3, -6, 0, -4, 0, 0, 5, -2, 1, 6, 8, 4, 3, -3, 2, -2, -5, 0, 0, -4, 0, -6, -3, 0, -9, 8, 2, 0, 2, 2, -5, 0, -6, 4, 0, 2, 0, 0, 3, 6, 2, -2

Offset

1,6

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q * f(-q) * f(-q^2) * f(-q^16)^3 / f(-q^2, -q^14) in powers of q where f() is a Ramanujan theta function.

Expansion of q * psi(-q) * psi(-q^2) * psi(-q^4) * f(-q^6, -q^10) in powers of q where psi(), f() are Ramanujan theta functions.

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

Example

G.f. = q - q^2 - q^3 - q^5 + 2*q^6 + 2*q^8 + q^11 - 2*q^12 - q^13 - 2*q^14 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^6, q^16] QPochhammer[ q^10, q^16] QPochhammer[ q^16]^2 QPochhammer[ q] QPochhammer[ q^4], {q, 0, n}];

Prog

(PARI) {a(n) = local(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\2, if( issquare( 16*k + 1, &m), (-1)^((m + 1) \ 8) * x^(2*k), 0), A), n))};
(Sage) A = CuspForms( Gamma1(16), 2, prec=80).basis(); A[0] - A[1];
(Magma) A := Basis( CuspForms( Gamma1(16), 2), 80); A[1] - A[2]; /* Michael Somos, Jan 08 2015 */

Crossrefs

Sequence in context: A248107 A352561 A035220 * A366533 A340683 A221645

Adjacent sequences: A227615 A227616 A227617 * A227619 A227620 A227621

Keyword

sign

Author

Michael Somos, Jul 17 2013

Status

approved