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!)
A259761 Expansion of (phi(x)^2 + phi(x^9)^2) / 2 in powers of x where phi() is a Ramanujan theta function. 4
1, 2, 2, 0, 2, 4, 0, 0, 2, 4, 4, 0, 0, 4, 0, 0, 2, 4, 4, 0, 4, 0, 0, 0, 0, 6, 4, 0, 0, 4, 0, 0, 2, 0, 4, 0, 4, 4, 0, 0, 4, 4, 0, 0, 0, 8, 0, 0, 0, 2, 6, 0, 4, 4, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 2, 8, 0, 0, 4, 0, 0, 0, 4, 4, 4, 0, 0, 0, 0, 0, 4, 4, 4, 0, 0, 8, 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

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

phi(x) = 1 + 2*Sum_{m=1..oo} x^(m^2). - N. J. A. Sloane, Jan 30 2017

Expansion of phi(x) * phi(x^9) + 2 * x^2 * chi(x^3)^2 * psi(-x^9)^2 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

a(n) = 2 * b(n) with a(0) = 1 and b() is multiplicative with b(2^e) = 1, b(3^e) = 1 + (-1)^e if e>0, b(p^e) = e+1 if p == 1, 5 (mod 12), (p^e) = (1 + (-1)^e)/2 if p == 7, 11 (mod 12).

a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0.

a(2*n) = a(n). a(9*n) = A004018(n). a(6*n + 4) = 2 * A122856(n).

EXAMPLE

G.f. = 1 + 2*x + 2*x^2 + 2*x^4 + 4*x^5 + 2*x^8 + 4*x^9 + 4*x^10 + 4*x^13 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x]^2 + EllipticTheta[ 3, 0, x^9]^2) / 2, {x, 0, n}];

a[ n_] := If[ n < 1, Boole[n == 0], 2 Times @@ ( Which[ # < 3, 1, # == 3, 1 + (-1)^#2, Mod[#, 12] < 6, #2 + 1, True, (1 + (-1)^#2) / 2 ] & @@@ FactorInteger[n])];

PROG

(PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 1 + (-1)^e, p%12>6, (1 + (-1)^e) / 2, e+1)))};

(MAGMA) A := Basis( ModularForms( Gamma1(36), 1), 87); A[1] + 2*A[2] + 2*A[3] + 2*A[5] + 4*A[6] + 2*A[9] + 4*A[10] + 4*A[11] + 4*A[14] + 2*A[17] + 4*A[18] + 4*A[19];

CROSSREFS

Cf. A004018, A122856.

Sequence in context: A141058 A102706 A105673 * A171933 A074823 A226177

Adjacent sequences:  A259758 A259759 A259760 * A259762 A259763 A259764

KEYWORD

nonn

AUTHOR

Michael Somos, Jul 04 2015

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 May 15 06:46 EDT 2021. Contains 343909 sequences. (Running on oeis4.)