A122854 Expansion of phi(q)^2*psi(q)^4 in powers of q where phi(),psi() are Ramanujan theta functions.

1, 8, 26, 48, 73, 120, 170, 208, 290, 360, 384, 528, 651, 656, 842, 960, 960, 1248, 1370, 1360, 1682, 1848, 1898, 2208, 2353, 2320, 2810, 3120, 2880, 3480, 3722, 3504, 4420, 4488, 4224, 5040, 5330, 5208, 5760, 6240, 5905, 6888, 7540, 6736, 7922, 8160, 7680

Offset: 0

Author

Michael Somos, Sep 14 2006

Keywords

• nonn
• easy

Comments

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

A050458(2n+1) = A050470(2n+1) = a(n).

Cf. A152584.

Cf. A000122, A000700, A010054, A121373.

Programs

Mathematica

a[n_]:= SeriesCoefficient[q^(-1/2)*(EllipticTheta[2, 0, q^(1/2)]^4 * EllipticTheta[3, 0, q]^2)/16, {q, 0, n}]; Table[A122854[n], {n, 0, 50}] (* G. C. Greubel, Jan 04 2018 *)

PARI

{a(n)= local(A, p, e, f); if(n<0, 0, n=2*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, f=(-1)^(p\2); (p^(2*e+2)-f^(e+1))/(p^2-f)))))}

PARI

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^18/(eta(x+A)^8*eta(x^4+A)^4), n))}

Formula

Expansion of q^(-1/2)eta(q^2)^18/(eta(q)^8*eta(q^4)^4) in powers of q.

Euler transform of period 4 sequence [ 8, -10, 8, -6, ...].

a(n)=b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = ((p^2)^(e+1)-1)/(p^2-1) if p == 1 (mod 4), b(p^e) = ((p^2)^(e+1)-(-1)^(e+1))/(p^2+1) if p == 3 (mod 4).

G.f.: Sum_{k>0 odd} k^2*x^k/(1+x^(2k)) = Product_{k>0} (1-x^(2k))^6*(1+x^k)^8/(1+x^(2k))^4.

Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^3/24 = 1.291928... (A152584). - Amiram Eldar, Dec 29 2023