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A033761 Product t2(q^d); d | 2, where t2 = theta2(q)/(2*q^(1/4)). 20
1, 1, 1, 2, 0, 1, 2, 1, 1, 1, 1, 0, 3, 1, 0, 2, 1, 1, 1, 0, 1, 3, 1, 2, 0, 0, 1, 2, 1, 0, 3, 1, 0, 2, 1, 1, 2, 0, 1, 0, 2, 1, 2, 1, 0, 3, 0, 1, 3, 0, 0, 2, 1, 0, 0, 1, 2, 4, 1, 1, 0, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 1, 2, 1, 0, 3, 0, 1, 4, 0, 1, 0, 1, 0, 2, 1, 1, 2, 0, 0, 2, 2, 1, 3, 0, 0, 2, 2, 1, 0, 2, 1, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

Also the number of representations of n as the sum of a triangular number and twice a triangular number. - James A. Sellers, Dec 21 2005

Also the number of positive odd solutions to equation x^2 + 2*y^2 = 8*n + 3. - Seiichi Manyama, May 28 2017

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000

M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Euler transform of period 4 sequence [1, 0, 1, -2, ...]. - Vladeta Jovovic, Sep 14 2004

Expansion of psi(q) * psi(q^2) in powers of q where psi() is a Ramanujan theta function.

Expansion of q^(-3/8) * eta(q^2) * eta^2(q^4) / eta(q) in powers of q. - Michael Somos, Jul 05 2006

Expansion of q^(-3/4) * (theta_2(q) * theta_2(q^2)) / 4 in powers of q^2. - Michael Somos, Jul 05 2006

Given g.f. A(x), then B(x) = x^3 * A(x^8) satisfies 0 = f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6) = u1^4*u6^2 + 3*u2^2*u3^4 - 4*u1*u2*u3*u6 * (u2^2 + 3*u6^2) - Michael Somos, Jul 05 2006

a(n) = A002325(8*n+3)/2. [Hirschhorn] - R. J. Mathar, Mar 23 2011

a(n) = A027414(8*n + 3). - Michael Somos, Nov 16 2011

G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082564. - Michael Somos, Jan 31 2015

EXAMPLE

G.f. = 1 + x + x^2 + 2*x^3 + x^5 + 2*x^6 + x^7 + x^8 + x^9 + x^10 + 3*x^12 + ...

G.f. = q^3 + q^11 + q^19 + 2*q^27 + q^43 + 2*q^51 + q^59 + q^67 + q^75 + q^83 + ...

MAPLE

sigmamr := proc(n, m, r) local a, d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d, m) = r then a := a+1 ; end if; end do: a; end proc:

A002325 := proc(n) sigmamr(n, 8, 1)+sigmamr(n, 8, 3)-sigmamr(n, 8, 5)-sigmamr(n, 8, 7) ; end proc:

A033761 := proc(n) A002325(8*n+3)/2 ; end proc:

seq(A033761(n), n=0..90) ; # R. J. Mathar, Mar 23 2011

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^2] / 4, {q, 0, 2 n + 3/4}]; (* Michael Somos, Nov 16 2011 *)

QP = QPochhammer; s = QP[q^2]*(QP[q^4]^2/QP[q]) + O[q]^105; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Nov 27 2015, adapted from PARI *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^2 / eta(x + A), n))}; /* Michael Somos, Jul 05 2006 */

(MAGMA) A := Basis( ModularForms( Gamma1(32), 1), 840); A[4] + A[12]; /* Michael Somos, Jan 31 2015 */

CROSSREFS

Cf. A027414, A097723, A033761-A033807, A082564.

Sequence in context: A085097 A117997 A079684 * A033805 A033797 A033793

Adjacent sequences:  A033758 A033759 A033760 * A033762 A033763 A033764

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, Sep 14 2004

STATUS

approved

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Last modified March 29 18:31 EDT 2020. Contains 333117 sequences. (Running on oeis4.)