A244553 Expansion of phi(q^2) * (phi(q) - phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

1, -1, 2, -1, 0, 2, 0, -1, 3, -4, 2, 2, 0, 0, 0, -1, 2, 1, 2, -4, 0, 2, 0, 2, 1, -4, 4, 0, 0, 0, 0, -1, 4, -2, 0, 1, 0, 2, 0, -4, 2, 0, 2, 2, 0, 0, 0, 2, 1, -5, 4, -4, 0, 4, 0, 0, 4, -4, 2, 0, 0, 0, 0, -1, 0, 4, 2, -2, 0, 0, 0, 1, 2, -4, 2, 2, 0, 0, 0, -4, 5

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q * f(-q, -q^7)^2 * phi(q^2) / psi(-q) = q * f(-q, -q^7)^2 * chi(q^2)^2 / chi(-q) in powers of q where phi(), psi(), f() are Ramanujan theta functions.

Euler transform of period 8 sequence [ -1, 2, 1, -2, 1, 2, -1, -2, ...].

Moebius transform is period 8 sequence [ 1, -2, 1, 0, -1, 2, -1, 0, ...].

a(2*n) = - A244554(n). a(2*n + 1) = A113411(n). a(8*n + 1) = A112603(n). a(8*n + 3) = 2 * A033761(n). a(8*n + 5) = a(8*n + 7) = 0.

Example

G.f. = q - q^2 + 2*q^3 - q^4 + 2*q^6 - q^8 + 3*q^9 - 4*q^10 + 2*q^11 + ...

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ {1, -2, 1, 0, -1, 2, -1, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2] (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}];

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, -2, 1, 0, -1, 2, -1][d%8 + 1]))};
(PARI) {a(n) = my(A, B); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); B = subst(A, x, x^2); polcoeff( B * (A - B) / 2, n))};
(Sage) A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[1] - A[2];
(Magma) A := Basis( ModularForms( Gamma1(8), 1), 33); A[2] - A[3];

Crossrefs

Cf. A033761, A113411, A112603, A244554.

Sequence in context: A065675 A348128 A328777 * A231167 A194313 A127476

Adjacent sequences: A244550 A244551 A244552 * A244554 A244555 A244556

Keyword

sign

Author

Michael Somos, Jun 30 2014

Status

approved