A244543 Expansion of phi(q^2) * (phi(q) + phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

1, 1, 3, 2, 3, 0, 2, 0, 3, 3, 4, 2, 2, 0, 0, 0, 3, 2, 5, 2, 4, 0, 2, 0, 2, 1, 4, 4, 0, 0, 0, 0, 3, 4, 6, 0, 5, 0, 2, 0, 4, 2, 0, 2, 2, 0, 0, 0, 2, 1, 7, 4, 4, 0, 4, 0, 0, 4, 4, 2, 0, 0, 0, 0, 3, 0, 4, 2, 6, 0, 0, 0, 5, 2, 4, 2, 2, 0, 0, 0, 4, 5, 6, 2, 0, 0, 2

Offset

0,3

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 = 0..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of f(-q^3, -q^5)^2 * phi(q^2) / psi(-q) = f(-q^3, -q^5)^2 * chi(q^2)^2 / chi(-q) in powers of q where phi(), psi(), chi(), 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) = A244540(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. = 1 + q + 3*q^2 + 2*q^3 + 3*q^4 + 2*q^6 + 3*q^8 + 3*q^9 + 4*q^10 + ...

Mathematica

a[ n_] := If[ n < 1, Boole[n == 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, n==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[0] + A[1] + 3*A[2];
(Magma) A := Basis( ModularForms( Gamma1(8), 1), 33); A[1] + A[2] + 3*A[3];

Crossrefs

Cf. A033761, A112603, A113411, A244540.

Sequence in context: A332497 A093347 A230409 * A283979 A134676 A286246

Adjacent sequences: A244540 A244541 A244542 * A244544 A244545 A244546

Keyword

nonn

Author

Michael Somos, Jun 29 2014

Status

approved