A204386 - OEIS

%I #24 Jan 22 2024 00:11:40

%S 1,12,28,96,126,336,344,768,757,1512,1332,2688,2198,4128,3528,6144,

%T 4914,9084,6860,12096,9632,15984,12168,21504,15751,26376,20440,33024,

%U 24390,42336,29792,49152,37296,58968,43344,72672,50654,82320,61544,96768

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

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

%H G. C. Greubel, <a href="/A204386/b204386.txt">Table of n, a(n) for n = 1..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.

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

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

%F 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).

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

%F Convolution of this sequence with A004018 is A050468.

%F From _Amiram Eldar_, Sep 12 2023: (Start)

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

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

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

%t 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 *)

%t 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 *)

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

%o (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))}

%Y Cf. A004018, A007331, A045823, A050468.

%Y Cf. A000122, A000700, A010054, A121373.

%K nonn,easy,mult

%O 1,2

%A _Michael Somos_, Jan 15 2012