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A033761 Product t2(q^d); d | 2, where t2 = theta2(q)/(2*q^(1/4)). 22
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
R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293 (see p. 285).
Iana I. Anguelova, The two bosonizations of the CKP hierarchy: overview and character identities, arXiv:1708.04992 [math-ph], 2017.
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
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
From Peter Bala, Jan 07 2021: (Start)
G.f.: A(x) = Sum_{n = -oo..oo} x^n/(1 - x^(8*n + 3)). See Agarwal, p. 285, equation 6.19.
A(x^2) = Sum_{n = -oo..oo} x^(2*n)/(1 - x^(8*n + 3)). Cf. A121444. (End)
A(q^2) = (1/2)*Sum_{k >= 0} q^k/(1 + q^(4*k+3)) + (1/2)*Sum_{k >= 0} q^(3*k)/(1 + q^(4*k+1)) - set z = 1 and replace q with q^2 in Anguelova, equation 3.35. - Peter Bala, Mar 03 2021
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-François 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
Sequence in context: A117997 A079684 A358138 * A033805 A033797 A033793
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Sep 14 2004
STATUS
approved

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Last modified June 18 02:11 EDT 2024. Contains 373468 sequences. (Running on oeis4.)