A122855 Expansion of (phi(q^3)phi(q^5) + phi(q)phi(q^15))/2 in powers of q where phi(q) is a Ramanujan theta function.

1, 1, 0, 1, 1, 1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 1, 3, 2, 0, 2, 1, 0, 0, 2, 2, 1, 0, 1, 0, 0, 0, 2, 4, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 3, 1, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 1, 2, 0, 0, 5, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 3, 1, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 4, 0, 0, 0, 1, 0, 0, 0, 0

Offset

0,9

Comments

Ramanujan theta functions: f(q) := Product_{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} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Alexander Berkovich and Hamza Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, The Ramanujan Journal, Vol. 20 (2009), pp. 375-408; arXiv preprint, arXiv:math/0611300 [math.NT], 2006-2007.

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Expansion of (eta(q^2)^2*eta(q^6)eta(q^10)eta(q^30)^2)/ (eta(q)eta(q^4)eta(q^15)eta(q^60)) in powers of q.

a(n) is multiplicative with a(2^e) = |e-1|, a(3^e)=a(5^e)=1, a(p^e) = e+1 if p == 1, 2, 4, 8 (mod 15), a(p^e) = (1+(-1)^e)/2 if p == 7, 11, 13, 14 (mod 15).

Euler transform of period 60 sequence [ 1, -1, 1, 0, 1, -2, 1, 0, 1, -2, 1, -1, 1, -1, 2, 0, 1, -2, 1, -1, 1, -1, 1, -1, 1, -1, 1, 0, 1, -4, 1, 0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, 1, 0, 2, -1, 1, -1, 1, -2, 1, 0, 1, -2, 1, 0, 1, -1, 1, -2, ...].

Moebius transform is period 60 sequence [ 1, -1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, -1, -1, 0, 1, 1, 0, -1, 0, 0, -1, -1, 0, 0, -1, 0, -1, -1, 0, -1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, 1, -1, 0, ...].

a(15n+7) = a(15n+11) = a(15n+13) = a(15n+14) = 0.

a(3n) = a(5n) = a(n).

G.f.: 1 + Sum_{k>0} Kronecker(-15,k) x^k/(1-(-x)^k).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(15) = 0.811155... . - Amiram Eldar, Nov 24 2023

Example

1 + q + q^3 + q^4 + q^5 + 2*q^8 + q^9 + q^12 + q^15 + ...

Mathematica

a[0] = 1; a[n_] := DivisorSum[n, KroneckerSymbol[-15, #]*(-1)^Boole[Mod[#, 4] == 2]&]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 07 2015, adapted from PARI *)

Prog

(PARI) {a(n)=if(n<1, n==0, sumdiv(n, d, kronecker(-15, d)*(-1)^(d%4==2)))}
(PARI) {a(n)= local(A, p, e); if(n<1, n==0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, e-1, if(p<7, 1, if(p%15==2^valuation(p%15, 2), e+1, 1-e%2))))))}
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2*eta(x^6+A)*eta(x^10+A)*eta(x^30+A)^2/ (eta(x+A)*eta(x^4+A)*eta(x^15+A)*eta(x^60+A)), n))}

Crossrefs

A035175(n) = a(4n).

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A359814 A359815 A260649 * A140727 A140728 A254110

Adjacent sequences: A122852 A122853 A122854 * A122856 A122857 A122858

Keyword

nonn,easy,mult

Author

Michael Somos, Sep 14 2006

Status

approved