A226240 Expansion of phi(q^4) * phi(q^8) + 2 * q *phi(q^2) * psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.

1, 2, 0, 4, 2, 0, 0, 0, 2, 6, 0, 4, 4, 0, 0, 0, 2, 4, 0, 4, 0, 0, 0, 0, 4, 2, 0, 8, 0, 0, 0, 0, 2, 8, 0, 0, 6, 0, 0, 0, 0, 4, 0, 4, 4, 0, 0, 0, 4, 2, 0, 8, 0, 0, 0, 0, 0, 8, 0, 4, 0, 0, 0, 0, 2, 0, 0, 4, 4, 0, 0, 0, 6, 4, 0, 4, 4, 0, 0, 0, 0, 10, 0, 4, 0, 0

Offset: 0

Author

Michael Somos, Jun 01 2013

Keywords

• nonn
• easy

Comments

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

Examples

			G.f. = 1 + 2*q + 4*q^3 + 2*q^4 + 2*q^8 + 6*q^9 + 4*q^11 + 4*q^12 + 2*q^16 + ...
		

Links

• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A033715, A113411.

Cf. A000122, A000700, A010054, A121373.

Programs

Magma

A := Basis( ModularForms( Gamma1(32), 1), 87); A[1] + 2*A[2] + 4*A[4] + 2*A[5] + 2*A[9] + 6*A[10] + 4*A[12] + 4*A[13] + 4*A[16]; /* Michael Somos, Jun 18 2014 */

Mathematica

a[ n_] := SeriesCoefficient[ 1 + 2 Sum[ Boole[ 2 != Mod[ k, 4]] q^k (1 + q^(2 k)) / (1 + q^(4 k)), {k, n}], {q, 0, n}];

PARI

{a(n) = if( n<1, n==0, 2 * (n%4 != 2) * sumdiv( n, d, kronecker( -2, d)))};

PARI

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 2 * prod( k= 1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, e>1, if( p%8<5, e+1, (1 + (-1)^e) / 2)))))};

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^8 + A) * eta(x^16 + A))^3 / (eta(x^4 + A) * eta(x^32 + A))^2 + 2 * x * eta(x^4 + A)^5 * eta(x^16 + A)^2 / (eta(x^2 + A)^2 * eta(x^8 + A)^3), n))};

Formula

Expansion of phi(q) * phi(q^8) + 4 * q^3 * psi(q^8) * psi(q^16) in powers of q where phi(), psi() are Ramanujan theta functions.

a(n) = 2 * b(n) where b(n) is multiplicative and b(2) = 0, b(2^e) = 1 if e>1, b(p^e) = e+1 if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 if p == 5, 7 (mod 8).

G.f.: 1 + 2 * Sum_{k>0 & k !=2 (mod 4)} q^k * (1 + q^(2*k)) / (1 + q^(4*k)).

a(4*n + 2) = 0. a(2*n + 1) = 2 * A113411(n). a(4*n) = A033715(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3*Pi/(4*sqrt(2)) = 1.6660811... . - Amiram Eldar, Dec 29 2023