login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A226240 Expansion of phi(q^4) * phi(q^8) + 2 * q *phi(q^2) * psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions. 2
1, 2, 0, 4, 2, 0, 0, 0, 2, 6, 0, 4, 4, 0, 0, 0, 2, 4, 0, 4, 0, 0, 0, 0, 4, 2, 0, 8, 0, 0, 0, 0, 2, 8, 0, 0, 6, 0, 0, 0, 0, 4, 0, 4, 4, 0, 0, 0, 4, 2, 0, 8, 0, 0, 0, 0, 0, 8, 0, 4, 0, 0, 0, 0, 2, 0, 0, 4, 4, 0, 0, 0, 6, 4, 0, 4, 4, 0, 0, 0, 0, 10, 0, 4, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

Table of n, a(n) for n=0..85.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of phi(q) * phi(q^8) + 4 * q^3 * psi(q^8) * psi(q^16) in powers of q where phi(), psi() are Ramanujan theta functions.

a(n) = 2 * b(n) where b(n) is multiplicative and b(2) = 0, b(2^e) = 1 if e>1, b(p^e) = e+1 if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 if p == 5, 7 (mod 8).

G.f.: 1 + 2 * Sum_{k>0 & k !=2 (mod 4)} q^k * (1 + q^(2*k)) / (1 + q^(4*k)).

a(4*n + 2) = 0. a(2*n + 1) = 2 * A113411(n). a(4*n) = A033715(n).

EXAMPLE

G.f. = 1 + 2*q + 4*q^3 + 2*q^4 + 2*q^8 + 6*q^9 + 4*q^11 + 4*q^12 + 2*q^16 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ 1 + 2 Sum[ Boole[ 2 != Mod[ k, 4]] q^k (1 + q^(2 k)) / (1 + q^(4 k)), {k, n}], {q, 0, n}];

PROG

(PARI) {a(n) = if( n<1, n==0, 2 * (n%4 != 2) * sumdiv( n, d, kronecker( -2, d)))};

(PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 2 * prod( k= 1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, e>1, if( p%8<5, e+1, (1 + (-1)^e) / 2)))))};

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^8 + A) * eta(x^16 + A))^3 / (eta(x^4 + A) * eta(x^32 + A))^2 + 2 * x * eta(x^4 + A)^5 * eta(x^16 + A)^2 / (eta(x^2 + A)^2 * eta(x^8 + A)^3), n))};

(MAGMA) A := Basis( ModularForms( Gamma1(32), 1), 87); A[1] + 2*A[2] + 4*A[4] + 2*A[5] + 2*A[9] + 6*A[10] + 4*A[12] + 4*A[13] + 4*A[16]; /* Michael Somos, Jun 18 2014 */

CROSSREFS

Cf. A033715, A113411.

Sequence in context: A305371 A177256 A199891 * A109468 A319690 A185879

Adjacent sequences:  A226237 A226238 A226239 * A226241 A226242 A226243

KEYWORD

nonn

AUTHOR

Michael Somos, Jun 01 2013

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 5 22:45 EDT 2020. Contains 333260 sequences. (Running on oeis4.)