A244544 Expansion of (phi(q) + phi(q^2))^2 / 4 in powers of q where phi() is a Ramanujan theta function.

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

Offset: 0

Author

Michael Somos, Jun 29 2014

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 2*q + 3*q^2 + 2*q^3 + 3*q^4 + 2*q^5 + 2*q^6 + 3*q^8 + 4*q^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A033761, A053692, A093709, A244540.

Programs

Magma

A := Basis( ModularForms( Gamma1(8), 1), 33); A[1] + 2*A[2] + 3*A[3];

Mathematica

a[ n_] := If[ n < 1, Boole[n == 0], Sum[ {2, 1, 0, 0, 0, -1, -2, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2])^2 / 4, {q, 0, n}];

PARI

{a(n) = if( n<1, n==0, sumdiv(n, d, [0, 2, 1, 0, 0, 0, -1, -2][d%8 + 1]))};

PARI

{a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( (A + subst(A, x, x^2))^2 / 4, n))};

Sage

A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[0] + 2*A[1] + 3*A[2];

Formula

Expansion of f(-q^3, -q^5)^4 / psi(-q)^2 in powers of q where phi(), f() are Ramanujan theta functions.

Euler transform of period 8 sequence [ 2, 0, -2, 2, -2, 0, 2, -2, ...].

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

Convolution square of A093709.

a(2*n) = A244540(n). a(8*n + 3) = 2*A033761(n). a(8*n + 5) = 2*A053692(n). a(8*n + 7) = 0.