login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 7
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 (list; graph; refs; listen; history; text; internal format)
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

A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, arXiv:math/0611300 [math.NT], 2006-2007.

M. Somos, Introduction to Ramanujan theta functions

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).

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-Fran├žois 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).

Sequence in context: A060398 A253242 A260649 * A140727 A140728 A254110

Adjacent sequences:  A122852 A122853 A122854 * A122856 A122857 A122858

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Sep 14 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 22 04:29 EDT 2019. Contains 322329 sequences. (Running on oeis4.)