A204386 Expansion of (theta_2(q)^8 + 4 * theta_2(q^2)^8) / 256 in powers of q^2.

1, 12, 28, 96, 126, 336, 344, 768, 757, 1512, 1332, 2688, 2198, 4128, 3528, 6144, 4914, 9084, 6860, 12096, 9632, 15984, 12168, 21504, 15751, 26376, 20440, 33024, 24390, 42336, 29792, 49152, 37296, 58968, 43344, 72672, 50654, 82320, 61544, 96768

Offset: 1

Author

Michael Somos, Jan 15 2012

Keywords

• nonn
• easy
• mult

Comments

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

Examples

			x + 12*x^2 + 28*x^3 + 96*x^4 + 126*x^5 + 336*x^6 + 344*x^7 + 768*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Michael Somos, Introduction to Ramanujan theta functions.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A004018, A007331, A045823, A050468.

Cf. A000122, A000700, A010054, A121373.

Programs

Mathematica

a[n_]:= SeriesCoefficient[(EllipticTheta[2, 0, q^(1/2)]^8 + 4*EllipticTheta[2, 0, q]^8)/256, {q, 0, n}];  Table[a[n], {n,1,50}] (* G. C. Greubel, Apr 13 2018 *)
CoefficientList[Series[(EllipticTheta[2,0,q^(1/2)]^8 +4*EllipticTheta[2, 0, q]^8)/ 256, {q, 0, 50}], q] (* Vaclav Kotesovec, Apr 13 2018 *)

PARI

{a(n) = if( n<1, 0, if( n%2, sigma( n, 3), 12 * sumdiv( n/2, d, (n/2/d%2) * d^3)))}

PARI

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A))^8 + 4 * x * (eta(x^4 + A)^2 / eta(x^2 + A))^8, n))}

Formula

Expansion of x * psi(x)^8 + 4 * x^2 * psi(x^2)^8 in powers of x where psi() is a Ramanujan theta function.

Expansion of (eta(q^2)^2 / eta(q))^8 + 4 * (eta(q^4)^2 / eta(q^2))^8 in powers of q.

a(n) is multiplicative with a(2^e) = 3/2 * 8^e if e>0, a(p^e) = ((p^3) ^ (e+1) - 1) / (p^3 - 1).

a(2*n + 1) = A045823(n). a(2*n) = 12 * A007331(n).

Convolution of this sequence with A004018 is A050468.

From Amiram Eldar, Sep 12 2023: (Start)

Dirichlet g.f.: (1 + 1/2^(s-2)) * (1 - 1/2^s) * zeta(s-3) * zeta(s).

Sum_{k=1..n} a(k) ~ c * n^4, where c = 5*Pi^4/1536 = 0.317086... . (End)