A133692 -id:A133692 - 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

A113411 Excess of number of divisors of 2n+1 of form 8k+1, 8k+3 over those of form 8k+5, 8k+7.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Oct 29 2005

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Bisection of A002325. Number of ways to write n as a sum of a square plus four times a triangular number [Hirschhorn]. - R. J. Mathar, Mar 23 2011

			1 + 2*x + 3*x^4 + 2*x^5 + 2*x^8 + 2*x^9 + x^12 + 4*x^13 + 4*x^16 + ...
q + 2*q^3 + 3*q^9 + 2*q^11 + 2*q^17 + 2*q^19 + q^25 + 4*q^27 + 4*q^33 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 82, Eq. (32.55).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael 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, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033761, A093954, A112603, A133692.

Cf. A000122, A000700, A010054, A121373.

a[n_] := DivisorSum[2n+1, Switch[Mod[#, 8], 1|3, 1, 5|7, -1]&]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 04 2015 *)

a(n) = if( n<0, 0, n = 2*n + 1; sumdiv(n, d, (-1)^(d%8>3)))

a(n) = local(n1); if( n<0, 0, n1 = sqrtint(n); polcoeff( sum(k=1,n1, 2*x^k^2, 1 + x*O(x^n)) * sum(k=0,n1, x^(2*k^2 + 2*k)), n))

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

a(n) = local(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod(k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p==2, 0, if( abs(p%8-6)==1, (1+(-1)^e)/2, e+1)))))

Expansion of phi(q) * psi(q^4) in powers of q where psi(), phi() are Ramanujan theta functions.
Expansion of q^(-1) * (eta(q^4)^5 * eta(q^16)^2) / (eta(q^2)^2 * eta(q^8)^3) in powers of q^2.
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = e+1 if p == 1, 3 (mod 8), b(p^e) = (1+(-1)^e)/2 if p == 5, 7 (mod 8).
Euler transform of period 8 sequence [ 2, -3, 2, 0, 2, -3, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A133692. - Michael Somos, Mar 16 2011
G.f.: (Sum_{k} x^k^2) * (Sum_{k>=0} x^(2*k^2 + 2*k)).
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>=0} F(x^(2*k + 1), x^(3*(2*k + 1))) where F(x, y) = (x + y) / (1 + x*y).
a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A112603(n). a(4*n + 1) = 2 * A033761(n).
From Peter Bala, Jan 07 2021: (Start)
Conjectural g.f.s: A(x) = Sum_{n >= 0} (-1)^(n*(n-1)/2)*x^n/(1 - x^(2*n+1)).
A(x) = Sum_{n = -oo..oo} (-1)^n*x^(2*n)/(1 - x^(4*n+1)) = Sum_{n = -oo..oo} (-1)^n*x^(2*n+1)/(1 - x^(4*n+3)). (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(2)) = 1.1107207... (A093954). - Amiram Eldar, Dec 28 2023

A082564 Expansion of eta(q)^2 * eta(q^2) / eta(q^4) in powers of q.

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

Benoit Cloitre, May 05 2003

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. - Michael Somos, Dec 15 2011
Absolute values appear to give A033715 = 2*A002325.
Denoted by a_4(n) in Kassel and Reutenauer 2015. - Michael Somos, Jun 04 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 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
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 , see page 31 7.2(c). [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, see p. 13 paragraph 3.3.3.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002479, A033715, A033761, A125095, A129134, A133692, A258747, A258764.

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

{a(n) = if( n<1, n==0, 2 * (-1)^((n+1) \ 2) * sumdiv( n, d, kronecker( -2, d)))}; /* Michael Somos, Mar 30 2007 */

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

Expansion of phi(-q) * phi(-q^2) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 30 2007
Euler transform of period 4 sequence [ -2, -3, -2, -2, ...]. - Michael Somos, Mar 30 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(11/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033761. - Michael Somos, Aug 29 2014
G.f.: Product_{k>0} (1 - x^k)^2 / (1 + x^(2*k)). - Michael Somos, Mar 30 2007
a(n) = -2 * A129134(n) unless n=0. - Michael Somos, Mar 30 2007
a(n) = (-1)^floor( (n+1)/2 ) * A033715(n). - Michael Somos, Aug 29 2014
a(2*n) = A133692(n). a(2*n + 1) = -2 * A125095(n). - Michael Somos, Aug 29 2014
a(3*n + 1) = -2 * A258747(n). a(3*n + 2) = -2 * A258764(n). - Michael Somos, Jun 09 2015

A133690 Expansion of (phi(-q) * phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 8, -16, 24, -24, 32, -32, 24, -52, 48, -48, 96, -56, 64, -96, 24, -72, 104, -80, 144, -128, 96, -96, 96, -124, 112, -160, 192, -120, 192, -128, 24, -192, 144, -192, 312, -152, 160, -224, 144, -168, 256, -176, 288, -312, 192, -192, 96, -228, 248, -288

Offset: 0

Michael Somos, Sep 20 2007

sign

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

			G.f. = 1 - 4*q + 8*q^2 - 16*q^3 + 24*q^4 - 24*q^5 + 32*q^6 - 32*q^7 + 24*q^8 - ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A004011, A008438, A097057, A097723, A112610, A133657, A133692.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -4 Which[ OddQ[n], DivisorSigma[ 1, n], Mod[n, 4] > 0, -2 DivisorSigma[1, n/2], True, -6 DivisorSum[n/4, # Mod[#, 2] &]]]; (* Michael Somos, Oct 30 2015 *)

{a(n) = if( n<1, n==0, -4 * if( n%2, sigma(n), n%4, -2 * sigma(n/2), -6 * sumdiv( n/4, d, (d%2)*d )))};

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

Expansion of (eta(q)^2 * eta(q^4)^5 / (eta(q^2)^3 * eta(q^8)^2))^2 in powers of q.
Euler transform of period 8 sequence [ -4, 2, -4, -8, -4, 2, -4, -4, ...].
a(n) = -4 * b(n) where b() is multiplicative with b(2) = -2, b(2^e) = -6 if e>1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133657.
G.f.: ( Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k))^3 / (1 + x^(4*k))^2 )^2.
a(n) = (-1)^n * A097057(n). Convolution square of A133692.
a(2*n) = 8 * A046897(n) unless n=0. a(2*n + 1) = A008438(n). a(4*n) = A004011(n). a(4*n + 1) = -4 * A112610(n). a(4*n + 3) = -16 * A097723(n).

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).

A133693 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

Offset: 1

Michael Somos, Sep 20 2007

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..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002325, A002479, A113411, A133692.

a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, KroneckerSymbol[ -2, #] &]]; (* Michael Somos, Oct 30 2015 *)

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

Expansion of (1 - eta(q)^2 * eta(q^4)^5 / (eta(q^2)^3 * eta(q^8)^2)) / 2 in powers of q.
Moebius transform is period 16 sequence [ 1, -2, 1, 0, -1, -2, -1, 0, 1, 2, 1, 0, -1, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -1 if e>0, a(p^e) = (1 + (-1)^e) / 2 if p == 5, 7 (mod 8), a(p^e) = e + 1 if p == 1, 3 (mod 8).
a(8*n + 5) = a(8*n + 7) = 0. A133692(n) = -2 * a(n) unless n=0. a(n) = -(-1)^n * A002325(n). a(2*n + 1) = A113411(n).

cp's OEIS Frontend

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

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A113411 Excess of number of divisors of 2n+1 of form 8k+1, 8k+3 over those of form 8k+5, 8k+7.

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A082564 Expansion of eta(q)^2 * eta(q^2) / eta(q^4) in powers of q.

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A133690 Expansion of (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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A133693 Expansion of (1 - phi(-q) * phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

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