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

0,2

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.

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

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 *)

Prog

(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))}

Crossrefs

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

Cf. A152584.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A031085 A224481 A031307 * A138502 A143894 A126176

Adjacent sequences: A122851 A122852 A122853 * A122855 A122856 A122857

Keyword

nonn,easy

Author

Michael Somos, Sep 14 2006

Status

approved