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

0,2

Comments

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

Links

Table of n, a(n) for n=0..85.

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

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

Example

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 + ...

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}];

Prog

(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))};
(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 */

Crossrefs

Cf. A033715, A113411.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A199891 A339417 A351558 * A109468 A331032 A319690

Adjacent sequences: A226237 A226238 A226239 * A226241 A226242 A226243

Keyword

nonn,easy,changed

Author

Michael Somos, Jun 01 2013

Status

approved