A255647 Expansion of (phi(q) * phi(q^22) + phi(q^2) * phi(q^11)) / 2 in powers of q where phi() is a Ramanujan theta function.

1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 1, 2, 0, 0, 2, 0, 2, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 1, 0, 2, 2, 0, 1, 1, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2

Offset

0,14

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, 2019.

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

Formula

a(n) is multiplicative with a(p^e) = 1 if p = 2 or 11, a(p^e) = e + 1 if Kronecker(-22, p) = +1, a(p^e) = (1 + (-1)^e)/2 if Kronecker(-22, p) = -1, and with a(0) = 1.

G.f. is a period 1 Fourier series which satisfies f(-1 / (88 t)) = 88^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

G.f.: 1 + Sum_{k>0} x^k / (1 - x^k) * Kronecker(-22, k).

a(n) = A035168(n) unless n = 0.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(22) = 0.669789... . - Amiram Eldar, Nov 23 2023

Example

G.f. = 1 + q + q^2 + q^4 + q^8 + q^9 + q^11 + 2*q^13 + q^16 + q^18 + 2*q^19 + ...

Mathematica

a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, KroneckerSymbol[ -22, #] &]];
a[ n_] := If[ n < 1, Boole[n == 0], Sum[ KroneckerSymbol[ -22, d], { d, Divisors[ n]}]];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^22] + EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^11]) / 2, {q, 0, n}];

Prog

(PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, kronecker( -22, d)))};
(PARI) {a(n) = if( n<1, n==0, direuler(p=2, n, 1 / ((1 - X) * (1 - kronecker(-22, p) * X)))[n])};
(PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2 || p==11, 1, kronecker( -22, p) == 1, e+1, 1-e%2)))};

Crossrefs

Cf. A035168.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A334158 A116357 A035168 * A119241 A001878 A056558

Adjacent sequences: A255644 A255645 A255646 * A255648 A255649 A255650

Keyword

nonn,easy,mult

Author

Michael Somos, May 05 2015

Status

approved