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

0,2

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..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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.

Example

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 + ...

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}];

Prog

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

Crossrefs

Cf. A033761, A053692, A093709, A244540.

Sequence in context: A214323 A321865 A353526 * A159580 A121549 A244228

Adjacent sequences: A244541 A244542 A244543 * A244545 A244546 A244547

Keyword

nonn

Author

Michael Somos, Jun 29 2014

Status

approved