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

Author

Michael Somos, Jun 11 2015

Keywords

• nonn
• easy

Comments

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

Examples

			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 + ...
		

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.

Crossrefs

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

Programs

GAP

sequence := List([1..10^5],n->Sigma(8*n-1)/8); # Muniru A Asiru, Dec 31 2017

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}];

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))};

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