A258835 Expansion of psi(x)^3 * psi(x^4) in powers of x where psi() is a Ramanujan theta function.

1, 3, 3, 4, 7, 6, 9, 13, 9, 10, 15, 15, 13, 19, 18, 16, 30, 21, 19, 27, 21, 31, 31, 24, 25, 39, 33, 28, 48, 30, 35, 54, 33, 34, 52, 42, 45, 51, 39, 45, 55, 51, 50, 70, 45, 46, 78, 48, 54, 80, 57, 63, 78, 54, 55, 75, 84, 58, 79, 60, 61, 117, 63, 74, 87, 72, 81

Offset

0,2

Comments

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

Links

Muniru A Asiru, Table of n, a(n) for n = 0..20000 (first 1000 terms from G. C. Greubel).

Michael Somos, Introduction to Ramanujan theta functions, 2019.

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

Formula

Expansion of q^(-7/8) * eta(q^2)^6 * eta(q^8)^2 / (eta(q)^3 * eta(q^4)) in powers of q.

Euler transform of period 8 sequence [ 3, -3, 3, -2, 3, -3, 3, -4, ...].

G.f.: Product_{k>0} (1 - x^(2*k))^4 * (1 + x^k)^3 * (1 + x^(2*k)) * (1 + x^(4*k))^2.

-8 * a(n) = A121613(4*n + 3). a(n) = sigma(8*n + 7) / 8.

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/16 = 0.6168502... (A222068). - Amiram Eldar, Mar 28 2024

Example

G.f. = 1 + 3*x + 3*x^2 + 4*x^3 + 7*x^4 + 6*x^5 + 9*x^6 + 13*x^7 + 9*x^8 + ...
G.f. = q^7 + 3*q^15 + 3*q^23 + 4*q^31 + 7*q^39 + 6*q^47 + 9*q^55 + 13*q^63 + ...

Maple

with(numtheory): seq(sigma(8*n-1)/8, n=1..1000); # Muniru A Asiru, Dec 31 2017

Mathematica

a[ n_] := If[ n < 0, 0, DivisorSigma[ 1, 8 n + 7] / 8];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x]^3 EllipticTheta[ 2, 0, x^4] / (16 x^(7/4)), {x, 0, n}];

Prog

(PARI) {a(n) = if( n<0, 0, sigma(8*n + 7) / 8)};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^8 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)), n))};
(GAP) sequence := List([1..10^5], n->Sigma(8*n-1)/8); # Muniru A Asiru, Dec 31 2017

Crossrefs

Cf. A000122, A000700, A010054, A121373, A121613, A222068.

Sequence in context: A266025 A216626 A319526 * A007448 A155689 A051263

Adjacent sequences: A258832 A258833 A258834 * A258836 A258837 A258838

Keyword

nonn,easy

Author

Michael Somos, Jun 11 2015

Status

approved