A082564 -id:A082564 - cp's OEIS frontend

A033715 Number of integer solutions (x, y) to the equation x^2 + 2y^2 = n.

1, 2, 2, 4, 2, 0, 4, 0, 2, 6, 0, 4, 4, 0, 0, 0, 2, 4, 6, 4, 0, 0, 4, 0, 4, 2, 0, 8, 0, 0, 0, 0, 2, 8, 4, 0, 6, 0, 4, 0, 0, 4, 0, 4, 4, 0, 0, 0, 4, 2, 2, 8, 0, 0, 8, 0, 0, 8, 0, 4, 0, 0, 0, 0, 2, 0, 8, 4, 4, 0, 0, 0, 6, 4, 0, 4, 4, 0, 0, 0, 0, 10, 4, 4, 0, 0, 4, 0, 4, 4, 0, 0, 0, 0, 0, 0, 4, 4, 2, 12, 2, 0, 8, 0

Offset: 0

N. J. A. Sloane

Theta series of lattice C2 with Gram matrix [ 1, 0; 0, 2]. a(n) is nonzero if and only if n is in A002479. - Michael Somos, Dec 15 2011
Number 17 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Denoted by |a_4(n)| in Kassel and Reutenauer 2015. - Michael Somos, Jun 16 2015

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

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 114 Entry 8(iii).
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 1999, p. 102, eq. 9.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 19.
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.24).
J. W. L. Glaisher, Table of the excess of the number of (8k+1)- and (8k+3)-divisors of a number over the number of (8k+5)- and (8k+7)-divisors, Messenger Math., 31 (1901), 82-91.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 346.

John Cannon, Table of n, a(n) for n = 0..10000
George E. Andrews, Richard Lewis, and Zhi-Guo Liu, An identity relating a theta series to a sum of Lambert series, Bull. London Math. Soc., 33 (2001), 25-31.
Michael Gilleland, Some Self-Similar Integer Sequences.
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852, Table I.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002325, A002479, A028572, A033761, A082564, A112603, A113411, A133692, A139093, A226225, A226240, A242609, A245572, A247719.
Cf. A000122, A000700, A010054, A121373.
Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), this sequence (d=-8), A028609 (d=-11), A028641 (d=-19), A138811 (d=-43).

A := Basis( ModularForms( Gamma1(8), 1), 105); A[1] + 2*A[2] + 2*A[3]; /* Michael Somos, Aug 29 2014 */

d:=proc(r,m,n) local i,t1; t1:=0; for i from 1 to n do if n mod i = 0 and i-r mod m = 0 then t1:=t1+1; fi; od: t1; end; [seq(2*(d(1,8,n)+d(3,8,n)-d(5,8,n)-d(7,8,n)),n=1..120)];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2], {q, 0, n}]; (* Michael Somos, Sep 09 2012 *)
a[ n_] := If[ n < 1, Boole[ n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -2, #] &]]; (* Michael Somos, Aug 29 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ q^4])^3 / (QPochhammer[ q] QPochhammer[ q^8])^2, {q, 0, n}]; (* Michael Somos, Aug 29 2014 *)

{a(n) = if( n<1, n==0, 2 * (issquare(n) - issquare(2*n) + 2 * sum( i=1, sqrtint(n\2), issquare(n - 2*i^2))))};

{a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -2, d)))}; /* Michael Somos, Aug 23 2005 */

{a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 2], n)[n])}; /* Michael Somos, Aug 23 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^-2 * eta(x^2 + A)^3 * eta(x^4 + A)^3 * eta(x^8 + A)^-2, n))};

Q = DiagonalQuadraticForm(ZZ, [1,2]); Q.representation_number_list(104); # Peter Luschny, Jun 20 2014

Fine gives an explicit formula for a(n) in terms of the divisors of n.
Euler transform of period 8 sequence [ 2, -1, 2, -4, 2, -1, 2, -2, ...].
Expansion of (eta(q^2) * eta(q^4))^3 / (eta(q) * eta(q^8))^2 in powers of q.
Coefficients in expansion of Sum_{i,j=-inf..inf} q^(i^2 + 2*j^2).
G.f. = s(2)^3*s(4)^3/(s(1)^2*s(8)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
G.f.: 1 + 2 * Sum_{k>0} Kronecker(-2, n) * x^k / (1 - x^k) = 1 + 2 * Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(4*k)).
G.f.: theta_3(q) * theta_3(q^2) = Product_{k>0} (1 + x^(2*k)) * ((1 + x^k) * (1 - x^(2*k)) / (1 + x^(4*k)))^2.
From Michael Somos, Oct 23 2006: (Start)
Moebius transform is period 8 sequence [ 2, 0, 2, 0, -2, 0, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u1 - 3*u3) * (u1 - u2 - u3 + u6) - (u2 - 3*u6) * (u1 - 2*u2 - u3 + 2*u6). (End)
a(n) = 2 * A002325(n) unless n = 0.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 8^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 09 2012
From Michael Somos, Aug 29 2014: (Start)
Expansion of phi(q) * phi(q^2) in powers of q where phi() is a Ramanujan theta function.
a(2*n) = a(n). a(2*n + 1) = 2 * A113411(n). (End)
From Michael Somos, May 17 2015: (Start)
a(n) = A028572(4*n) = A133692(2*n) = A139093(8*n) = A226225(8*n) = A226240(4*n) = A242609(4*n) = A245572(4*n) / 3 = (-1)^floor((n + 1)/2) * A082564(n).
a(8*n + 5) = a(8*n + 7) = 0. a(8*n + 1) = 2 * A112603(n). a(8*n + 3) = 4 * A033761(n). (End)
a(0) = 1, a(n) = 2 * b(n) for n > 0, where b() is multiplicative with b(2^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). - Jianing Song, Sep 04 2018 [Corrected by Jeremy Lovejoy, Nov 12 2024]
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi/sqrt(2) = 2.221441... (A247719). - Amiram Eldar, Dec 16 2023

A033761 Product t2(q^d); d | 2, where t2 = theta2(q)/(2*q^(1/4)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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 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

			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 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
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.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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

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

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

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 *)

{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 */

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

More terms from Vladeta Jovovic, Sep 14 2004

A129134 Expansion of (1 - phi(-q) * phi(-q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, -2, -1, 0, 2, 0, -1, 3, 0, -2, -2, 0, 0, 0, -1, 2, 3, -2, 0, 0, 2, 0, -2, 1, 0, -4, 0, 0, 0, 0, -1, 4, 2, 0, -3, 0, 2, 0, 0, 2, 0, -2, -2, 0, 0, 0, -2, 1, 1, -4, 0, 0, 4, 0, 0, 4, 0, -2, 0, 0, 0, 0, -1, 0, 4, -2, -2, 0, 0, 0, -3, 2, 0, -2, -2, 0, 0, 0, 0, 5, 2, -2, 0, 0, 2, 0, -2, 2, 0, 0, 0, 0, 0, 0, -2, 2, 1, -6, -1, 0, 4, 0, 0, 0

Offset: 1

Michael Somos, Mar 30 2007

sign

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

For n nonzero, a(n) is nonzero if and only if n is in A002479.

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

G. C. Greubel, Table of n, a(n) for n = 1..5000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002325, A002479, A082564, A258747, A258764.

a[ n_] := If[ n < 1, 0, (-1)^Quotient[ n - 1, 2] DivisorSum[n, KroneckerSymbol[-2, #] &]]; (* Michael Somos, Jun 09 2015 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^2]) / 2 , {q, 0, n}]; (* Michael Somos, Jun 09 2015 *)
a[ n_] := SeriesCoefficient[ (1 - QPochhammer[ q]^2 QPochhammer[ q^2] / QPochhammer[ q^4]) / 2 , {q, 0, n}]; (* Michael Somos, Jun 09 2015 *)

{a(n) = if( n<1, 0, (-1)^((n-1)\2) * sumdiv(n, d, kronecker( -2, d)))};

{a(n) = local(A); if( n<1, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^2 * eta(x^2 + A) / eta(x^4 + A)) / 2, n))};

Expansion of (1 - eta(q)^2 * eta(q^2) / eta(q^4)) / 2 in powers of q.
G.f.: (1 - Product_{k>0} (1 - x^k)^2 / (1 + x^(2*k)) )/2.
a(n) = A002325(n) * (-1)^floor((n-1)/2). A082564(n) = -2 * a(n) unless n=0.
a(3*n + 1) = A258747(n). a(3*n + 2) = A258764(n). - Michael Somos, Jun 09 2015

A139093 Expansion of phi(q) * phi(-q^2) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, -2, -4, 2, 0, -4, 0, 2, 6, 0, -4, 4, 0, 0, 0, 2, 4, -6, -4, 0, 0, -4, 0, 4, 2, 0, -8, 0, 0, 0, 0, 2, 8, -4, 0, 6, 0, -4, 0, 0, 4, 0, -4, 4, 0, 0, 0, 4, 2, -2, -8, 0, 0, -8, 0, 0, 8, 0, -4, 0, 0, 0, 0, 2, 0, -8, -4, 4, 0, 0, 0, 6, 4, 0, -4, 4, 0, 0, 0, 0, 10, -4, -4, 0, 0, -4, 0, 4, 4, 0, 0, 0, 0, 0, 0, 4, 4, -2, -12, 2, 0, -8, 0

Offset: 0

Michael Somos, Apr 08 2008

sign

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

a(n) is nonzero if and only if n is in A002479.

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

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002479, A033715, A033761, A082564, A112603, A125095, A133692.

A := Basis( ModularForms( Gamma1(32), 1), 105); A[1] + 2*A[2] - 2*A[3] - 4*A[4] + 2*A[5] - 4*A[7] + 2*A[9] + 6*A[10] - 4*A[12] + 4*A[13] + 4*A[16]; /* Michael Somos, Aug 29 2014 */

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^2], {q, 0, n}]; (* Michael Somos, Aug 29 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^7 / (QPochhammer[ q]^2 QPochhammer[ q^4]^3), {q, 0, n}]; (* Michael Somos, Feb 18 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], 2 (-1)^Quotient[n, 2] Sum[ JacobiSymbol[ -2, d], {d, Divisors @ n}]]; (* Michael Somos, Feb 18 2015 *)

{a(n) = if( n<1, n==0, 2 * (-1)^(n\2) * sumdiv(n, d, kronecker( -2, d)))};

{a(n) = my(A); if ( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^3), n))};

Expansion of eta(q^2)^7 / (eta(q)^2 * eta(q^4)^3) in powers of q.
Euler transform of period 4 sequence [ 2, -5, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 8^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A112603.
G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k-1))^2 / (1 + x^(2*k)).
a(8*n + 5) = a(8*n + 7) = 0.
a(n) = (-1)^n * A082564(n). a(2*n) = A133692(n). a(2*n + 1) = 2 * A125095(n). a(4*n) = a(8*n) = A033715(n). a(8*n + 1) = 2 * A112603(n). a(8*n + 3) = -4 * A033761(n).

A258747 Expansion of chi(-x) * f(x^3) * f(-x^6) in powers of x where chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, 0, 0, 0, -1, -2, 2, 1, 0, 0, 2, 0, 0, -2, 0, 1, 0, 0, 0, 0, -1, -2, 0, 2, -2, 0, 2, 0, -2, 0, 0, 2, -1, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, 0, 0, 0, -2, 2, 1, -2, 0, 2, 0, 0, -4, 0, 2, -1, 0, 0, 0, 0, -2, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2

Offset: 0

Michael Somos, Jun 09 2015

sign

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

			G.f. = 1 - x - x^5 - 2*x^6 + 2*x^7 + x^8 + 2*x^11 - 2*x^14 + x^16 - x^21 + ...
G.f. = q - q^4 - q^16 - 2*q^19 + 2*q^22 + q^25 + 2*q^34 - 2*q^43 + q^49 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A082564, A129134, A255318, A255319, A257398.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^3] QPochhammer[ x^6], {x, 0, n}];
a[ n_] := If[ n < 0, 0, (-1)^Quotient[ 3 n, 2] DivisorSum[ 3 n + 1, KroneckerSymbol[-2, #] &]];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^4 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))};

{a(n) = if( n<0, 0, (-1)^(3*n\2) * sumdiv(3*n + 1, d, kronecker( -2, d)))};

Expansion of q^(-1/3) * eta(q) * eta(q^6)^4 / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -1, 0, 0, 0, -1, -3, -1, 0, 0, 0, -1, -2, ...].
G.f.: Product_{k>0} (1 + x^(3*k)) * (1 - x^(6*k))^2 / ( (1 + x^k) * (1 + x^(6*k)) ).
-2 * a(n) = A082564(3*n + 1). a(n) = A129134(3*n + 1).
a(4*n + 3) = 2 * A257402(n-1). a(8*n) = A257398(n). a(8*n + 2) = a(8*n + 4) = a(16*n + 3) = a(16*n + 15) = 0. a(16*n + 7) = 2 * A255318(n). a(16*n + 11) = 2 * A255319(n).

A258764 Expansion of chi(-x^2) * psi(-x^3)^2 in powers of x where psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 0, -1, -2, 0, 2, 0, 0, 0, 0, -1, 0, 2, 2, -2, 0, 1, 0, 0, -2, 0, 0, -2, 0, 0, 0, 0, -2, 2, 2, 0, 0, 1, 0, 0, -2, 0, 2, 0, 0, 0, 0, -1, -2, 2, 2, 0, 0, 2, 0, -2, 0, 0, 0, -2, 0, 0, 0, -2, -2, 0, 0, 0, 0, 2, 0, -1, 0, 0, 4, 0, 0, 0, 0, 0, -2, 0, 2, -2, 0, 3

Offset: 0

Michael Somos, Jun 09 2015

sign

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

			G.f. = 1 - x^2 - 2*x^3 + 2*x^5 - x^10 + 2*x^12 + 2*x^13 - 2*x^14 + x^16 + ...
G.f. = q^2 - q^8 - 2*q^11 + 2*q^17 - q^32 + 2*q^38 + 2*q^41 - 2*q^44 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A082564, A129134, A255317, A255318, A255319, A257399, A257402.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^4] QPochhammer[ x^3]^2 / QPochhammer[ x^6, x^12]^2, {x, 0, n}];
a[ n_] := If[ n < 0, 0, (-1)^Quotient[ n, 2] DivisorSum[ 3 n + 2, KroneckerSymbol[-2, #] &]];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2 / (eta(x^4 + A) * eta(x^6 + A)^2), n))};

{a(n) = if( n<0, 0, (-1)^(n\2) * sumdiv(3*n + 2, d, kronecker( -2, d)))};

Expansion of q^(-2/3) * eta(q^2) * eta(q^3)^2 * eta(q^12)^2 / (eta(q^4) * eta(q^6)^2) in powers of q.
Euler transform of period 12 sequence [ 0, -1, -2, 0, 0, -1, 0, 0, -2, -1, 0, -2, ...].
G.f.: Product_{k>0} (1 + x^(2*k)) * (1 - x^(3*k))^2 * (1 - x^(2*k) + x^(4*k))^2.
a(n) = A129134(3*n + 2). -2 * a(n) = A082564(3*n + 2).
a(4*n) = A257399(n). a(8*n + 3) = -2 * A255318(n). a(8*n + 5) = 2 * A255319(n). a(8*n + 6) = -2 * A257402(n-1). a(16*n) = A257398(n). a(16*n + 2) = - A257399(n). a(16*n + 12) = 2 * A255317(n).
a(8*n + 1) = a(8*n + 7) = a(16*n + 4) = a(16*n + 8) = 0.

cp's OEIS Frontend

A033715 Number of integer solutions (x, y) to the equation x^2 + 2y^2 = n.

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A033761 Product t2(q^d); d | 2, where t2 = theta2(q)/(2*q^(1/4)).

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A129134 Expansion of (1 - phi(-q) * phi(-q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

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A139093 Expansion of phi(q) * phi(-q^2) in powers of q where phi() is a Ramanujan theta function.

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A258747 Expansion of chi(-x) * f(x^3) * f(-x^6) in powers of x where chi(), f() are Ramanujan theta functions.

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A258764 Expansion of chi(-x^2) * psi(-x^3)^2 in powers of x where psi(), chi() are Ramanujan theta functions.

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