A111374 -id:A111374 - cp's OEIS frontend

A327851 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A111374.

1, 1, 2, 2, 4, 4, 6, 8, 12, 15, 19, 24, 30, 36, 47, 57, 74, 88, 112, 130, 160, 190, 232, 277, 333, 399, 471, 554, 656, 768, 908, 1060, 1256, 1452, 1702, 1968, 2294, 2646, 3068, 3549, 4093, 4710, 5418, 6211, 7121, 8138, 9331, 10625, 12150, 13817, 15749, 17858, 20290, 23000, 26054

Offset: 0

Seiichi Manyama, Sep 28 2019

nonn

a(n) > 0.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Convolution inverse of A327852.
Product_{k>=1} (1 - x^k)^(- Sum_{d|k} (b/d)), where (m/n) is the Kronecker symbol: this sequence (b=2), A107742 (b=4), A327716 (b=5).
Cf. A035185, A111374.

nmax = 60; CoefficientList[Series[Product[QPochhammer[x^(8*j - 3)] * QPochhammer[x^(8*j - 5)]/(QPochhammer[x^(8*j - 7)] * QPochhammer[x^(8*j - 1)]), {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 28 2019 *)

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1-x^k)^sumdiv(k, d, kronecker(2, d))))

G.f.: Product_{i>=1} Product_{j>=1} (1-x^(i*(8*j-3))) * (1-x^(i*(8*j-5))) / ((1-x^(i*(8*j-1))) * (1-x^(i*(8*j-7)))).

G.f.: Product_{k>=1} (1-x^k)^(-A035185(k)).

A092869 Series expansion of the Ramanujan-Goellnitz-Gordon continued fraction.

Original entry on oeis.org

1, -1, 0, 1, -1, 1, 0, -2, 2, -1, 0, 2, -3, 2, 0, -2, 4, -4, 0, 4, -6, 5, 0, -6, 9, -6, 0, 7, -12, 9, 0, -10, 16, -13, 0, 15, -22, 17, 0, -20, 29, -21, 0, 25, -38, 28, 0, -32, 50, -39, 0, 43, -64, 49, 0, -56, 82, -60, 0, 69, -105, 78, 0, -86, 132, -101, 0, 112, -166, 125, 0, -142, 208, -153, 0, 172, -258, 192, 0

Offset: 0

Michael Somos, Mar 07 2004; corrected Jun 09 2004

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Glaisher (1876) writes "XIII. tan(pi/16) = (e^(-pi/2) - e^(-3 pi/2) - e^(-15 pi/2) + e^(-21 pi/2) + e^(-45 pi/2) - &c.) / (1 - e^(-6 pi/2) - e^(-10 pi/2) + e^(-28 pi/2) + e^(-36 pi/2) - &c.), ..." where the numerator is q * f( -q^2, -q^14) and denominator is f( -q^6, -q^10) where q = e^(-pi/2). - Michael Somos, Jun 22 2012
Berndt writes "[...] v = q^(1/2) f(-q,-q^7) / f(-q^3,-q^5). Then v = q^(1/2) / (1 + q + q^2 / (1 + q^3 + q^4 / (1 + q^5 + q^6 / (1 + x^7 + ...)))). (1.1)". - Michael Somos, Jul 09 2012
Jacobi writes "(7.) (1 - sqrt(k')) / (1 + sqrt(k') + sqrt(2(1+k'))) = (q - q^3 - q^15 + q^21 + q^45 - q^55 - ...) / (1 - q^6 - q^10 + q^28 + q^36 - q^66 - ...)." - Michael Somos, Sep 11 2012

			G.f. = 1 - x + x^3 - x^4 + x^5 - 2*x^7 + 2*x^8 - x^9 + 2*x^11 - 3*x^12 + ...
G/f. = q - q^3 + q^7 - q^9 + q^11 - 2*q^15 + 2*q^17 - q^19 + 2*q^23 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag; see p. 221 Entry 1(ii), eq. (1.1).
J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), 111-112. see Eq. XIII
C. G. J. Jacobi, Über die Zur Numerischen Berechnung der Elliptischen Functionen Zweckmaessigsten Formeln, Crelle Bd. 26 (1843), 93-114 = Gesammelte Werke, Bd. 1, 1881, 343-368.

G. C. Greubel, Table of n, a(n) for n = 0..1000
S.-D. Chen and S.-S. Huang, On the series expansion of the Göllnitz-Gordon continued fraction, Internat. J. Number Theory, 1 (2005), 53-63.
B. Cho, J. K. Koo, and Y. K. Park, Arithmetic of the Ramanujan-Göllnitz-Gordon continued fraction, J. Number Theory, 129 (2009), 922-947.
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162; see Eqs. (9.2), (9.4).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000700, A003823, A083365, A091188, A111374, A226559.

A := Basis( ModularForms( Gamma1(16), 1/2), 159); LS := LaurentSeriesRing( RationalField()); A[3] / A[2]; /* Michael Somos, Aug 31 2018 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^8] QPochhammer[ x^7, x^8] /(QPochhammer[ x^3, x^8] QPochhammer[ x^5, x^8]), {x, 0, n}] (* Michael Somos, Aug 02 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^2] / (EllipticTheta[ 3, 0, x] + EllipticTheta[ 3, 0, x^2]), {x, 0, n + 1/2}] (* Michael Somos, Aug 02 2011 *)
a[ n_] := SeriesCoefficient[ Product[(1 - q^k)^KroneckerSymbol[ 8, k], {k, n}], {q, 0, n}] (* Michael Somos, Jul 08 2012 *)

{a(n) = local(A, u, v); if( n<0, 0, n = 2*n + 1; A = x; forstep( k=3, n, 2, u = A + x * O(x^k); v = subst(u, x, x^2); A -= x^k * polcoeff(u^2 - v + v*u^2 + v^2, k+1) / 2); polcoeff(A, n))}

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

{a(n) = local(A, A2); if( n<0, 0, A = eta(x^8 + x * O(x^n))^2 / eta(x^4 + x * O(x^n)); A2 = sum( k=1, sqrtint(n), x^k^2 + x^(2*k^2), 1 + x * O(x^n)); polcoeff(A / A2, n))}

{a(n) = local(A, A2); if( n<0, 0, A = x * O(x^n); A = eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3; A2 = subst(A, x, x^2); polcoeff( 2 * A^2 * A2^2 / (A^2 + A2), n))}

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-x^k)^kronecker(2, k))) \\ Seiichi Manyama, Sep 24 2019

Expansion of f(-x, -x^7) / f(-x^3, -x^5) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Aug 02 2011
Expansion of (phi(x) - phi(x^2)) / (2 * x * psi(x^4)) = 2 * psi(x^4) / (phi(x) + phi(x^2)) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 15 2006
Expansion of q^(-1) * (1 - sqrt(k')) / (1 + sqrt(k') + sqrt(2 * (1 + k'))) in powers of q^2 where k' is the complementary elliptic modulus. - Michael Somos, Sep 11 2012
Euler transform of period 8 sequence [-1, 0, 1, 0, 1, 0, -1, 0, ...].
G.f. A(x) satisfies both A(-x) * A(x) = A(x^2) and x * A(x)^2 = B(x * A(x^2)) where B(x) = x * (1 - x) / (1 + x).
Given g.f. A(x), then B(x) = x * A(x^2) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 - v + v^2 + v*u^2.
Given g.f. A(x), then B(x) = x * A(x^2) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = (1 - u*v) * (u + v)^3 - v * (1 + v^2) * (1 - u^4). - Michael Somos, Feb 15 2006
Given g.f. A(x), then B(x) = x * A(x^2) satisfies 0 = f(B(x), B(x^5)) where f(u, v) = (u - v) * (1 + u*v)^5 - u * (1 - u^4) * (1 + v^2) * (1 - 6*v^2 + v^4). - Michael Somos, Feb 15 2006
G.f.: Product_{k>=0} (1 - x^(8*k + 1)) * (1 - x^(8*k + 7)) / ((1 - x^(8*k + 3)) * (1 - x^(8*k + 5))).
G.f. = continued fraction 1/(1 + x + x^2/(1 + x^3 + x^4/(1 + x^5 + x^6/(1 + x^7 + ...)))). Convolution inverse of A111374.
a(2*n + 1) = -A226559(n). - Michael Somos, Jun 12 2013
a(4*n) = A083365(n). a(4*n + 2) = 0.
G.f. A(x) satisfies x*A(-x^2) = x*B(x^2)/C(x^2) = (F(x) - F(-x))/(F(x) + F(-x)), where B(x) is the g.f. of A069911, C(x) is the g.f. of A069910 and F(x) = Product_{k >= 0} 1 + x^(2*k+1) is the g.f. of A000700. - Peter Bala, Feb 07 2021

A069910 Expansion of Product_{i in A069908} 1/(1 - x^i).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 11, 12, 16, 18, 23, 26, 33, 37, 46, 52, 63, 72, 87, 98, 117, 133, 157, 178, 209, 236, 276, 312, 361, 408, 471, 530, 609, 686, 784, 881, 1004, 1126, 1279, 1433, 1621, 1814, 2048, 2286, 2574, 2871, 3223, 3590, 4022, 4472, 5000

Offset: 0

N. J. A. Sloane, May 05 2002

Number 39 of the 130 identities listed in Slater 1952.

Number of partitions of 2*n into distinct odd parts. - Vladeta Jovovic, May 08 2003

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 5*x^9 + ...
G.f. = q^-1 + q^95 + q^143 + 2*q^191 + 2*q^239 + 3*q^287 + 3*q^335 + ...

M. D. Hirschhorn, The Power of q, Springer, 2017. Chapter 19, Exercises p. 173.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
G. E. Andrews et al., q-Engel series expansions and Slater's identities, Quaestiones Math., 24 (2001), 403-416.
George E. Andrews, Jethro van Ekeren and Reimundo Heluani, The singular support of the Ising model, arXiv:2005.10769 [math.QA], 2020. See (1.4.2) p. 2.
T. Gannon, G. Hoehn, H. Yamauchi, et. al., VOA Unitary Minimal Model m=1, character.
M. D. Hirschhorn, Some partition theorems of the Rogers-Ramanujan type, J. Combin. Theory Ser. A 27 (1979), no. 1, 33-37. MR0541341 (80j:05010). See Theorem 4. [From _N. J. A. Sloane_, Mar 19 2012]
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Lucy Joan Slater, Further Identities of the Rogers-Ramanujan Type, Proc. London Math. Soc., Series 2, vol.s2-54, no.2, pp. 147-167, (1952).
Eric Weisstein's World of Mathematics, Jackson-Slater Identity
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el], 2012. - _N. J. A. Sloane_, Dec 25 2012

Cf. A000700, A027356, A035457, A069908, A069909, A069911, A122129, A122135.

a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*[0$2, 1$4, 0$5, 1$4, 0][irem(d, 16)+1],
      d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

max = 56; p = Product[1/(1-x^i), {i, Select[Range[max], MemberQ[{2, 3, 4, 5, 11, 12, 13, 14}, Mod[#, 16]]&]}]; s = Series[p, {x, 0, max}]; a[n_] := Coefficient[s, x, n]; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Apr 09 2014 *)
nmax=60; CoefficientList[Series[Product[(1-x^(8*k-1))*(1-x^(8*k-7))*(1-x^(8*k))*(1-x^(16*k-6))*(1-x^(16*k-10))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 04 2015 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^-{ 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0 }[[ Mod[k, 16] + 1]], {k, n}], {x, 0, n}]; (* Michael Somos, Apr 14 2016 *)

{a(n) = my(A); if( n<0,0, n=2*n; A = x * O(x^n); polcoeff( eta(-x + A) / eta(x^2 + A), n))}; /* Michael Somos, Apr 11 2004 */

N=66;  q='q+O('q^N);  S=1+sqrtint(N);
gf=sum(n=0, S, q^(2*n^2) / prod(k=1, 2*n, 1-q^k ) );
Vec(gf)  \\ Joerg Arndt, Apr 01 2014

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^-[ 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0][k%16 + 1]), n))}; /* Michael Somos, Apr 14 2016 */

Euler transform of period 16 sequence [0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, ...]. - Michael Somos, Apr 11 2004
G.f.: Sum_{n>=0} q^(2*n^2) / Product_{k=1..2*n} (1 - q^k). - Joerg Arndt, Apr 01 2014
a(n) ~ exp(sqrt(n/3)*Pi) / (2^(5/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 04 2015
Expansion of f(x^3, x^5) / f(-x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Apr 14 2016
a(n) = A000700(2*n).
a(n) = A027356(4n+1,2n+1). - Alois P. Heinz, Oct 28 2019
From Peter Bala, Feb 08 2021: (Start)
G.f.: A(x) = Product_{n >= 1} (1 + x^(4*n))^2*(1 + x^(4*n-2))*(1 + x^(8*n-3))*(1 + x^(8*n-5)).
The 2 X 2 matrix Product_{k >= 0} [1, x^(2*k+1); x^(2*k+1), 1] = [A(x^2), x*B(x^2); x*B(x)^2, A(x^2)], where B(x) is the g.f. of A069911.
A(x^2) + x*B(x^2) = A^2(-x) + x*B^2(-x) = Product_{k >= 0} 1 + x^(2*k+1), the g.f. of A000700.
A^2(x) + x*B^2(x) is the g.f. of A226622.
(A^2(x) + x*B^2(x))/(A^2(x) - x*B^2(x)) is the g.f. of A208850.
A^4(sqrt(x)) - x*B^4(sqrt(x)) is the g.f. of A029552.
A(x)*B(x) is the g.f. of A226635; A(-x)/B(-x) is the g.f. of A111374; B(-x)/A(-x) is the g.f. of A092869. (End)

A117000 a(n) = Sum_{d|n} Jacobi(2,d)*d.

Original entry on oeis.org

1, 1, -2, 1, -4, -2, 8, 1, 7, -4, -10, -2, -12, 8, 8, 1, 18, 7, -18, -4, -16, -10, 24, -2, 21, -12, -20, 8, -28, 8, 32, 1, 20, 18, -32, 7, -36, -18, 24, -4, 42, -16, -42, -10, -28, 24, 48, -2, 57, 21, -36, -12, -52, -20, 40, 8, 36, -28, -58, 8, -60, 32, 56, 1, 48, 20, -66, 18, -48, -32, 72, 7, 74, -36, -42, -18, -80, 24, 80, -4

Offset: 1

N. J. A. Sloane, Apr 15 2006

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

Inverse Mobius transform of the sequence n*A091337(n), n>=1. - R. J. Mathar, Jul 08 2011

			G.f. = q + q^2 - 2*q^3 + q^4 - 4*q^5 - 2*q^6 + 8*q^7 + q^8  + 7*q^9 - 4*q^10 - 10*q^11 + ...

Henry J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 323.
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.67).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Apart from signs, same as A113418.
Cf. A091337, A111374, A117001.
Cf. A000122, A000700, A010054, A121373.

with(numtheory); A117000:=proc(n) local d,t1,t2; t1:=0; t2:=0; for d from 1 to n do if n mod d = 0 then t1:=t1+jacobi(2,d)*d; fi; od: t1; end;

a[n_] := Sum[JacobiSymbol[2, d]*d, {d, Divisors[n]}]; a /@ Range[80] (* Jean-François Alcover, Jan 10 2014 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 4, 0, q]^2) / 2, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ (1 - QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^3 / QPochhammer[ q^8]^2) / 2, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)

{a(n)= if( n<1, 0, sumdiv(n, d, d * kronecker(2, d)))}; /* Michael Somos, Aug 08 2007 */

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 1, if( abs(p%8-4)==3, (p^(e+1)-1)/(p-1), ((-p)^(e+1)-1)/(-p-1))))))}; /* Michael Somos, Aug 08 2007 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^2 * eta(x^2 + A) * eta(x^4 + A)^3 / eta(x^8 + A)^2) / 2, n))}; /* Michael Somos, Aug 08 2007 */

G.f.: Sum_{k>0} x^k*(1+x^(2*k))*(1-4*x^(2*k)+x^(4*k))/(1+x^(4*k))^2. - Vladeta Jovovic, Apr 15 2006
From Michael Somos, Aug 08 2007: (Start)
Expansion of (1 - phi(q) * phi(q^2) * phi(-q)^2) / 2 in powers of q where phi() is a Ramanujan theta function.
a(n) is multiplicative with a(2^e) = 1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 7 (mod 8), ((-p)^(e+1) - 1) / (-p - 1) if p == 3, 5 (mod 8).
Given g.f. A(x), then B(x) = 1 - 2*A(x) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v^4 + u^2*v^2 + 2*u^2*w^2 + 2*u*v*w * (-u+2*v-2*w) - 2*u*v^3.
G.f.: Sum_{k>0} k * x^k / (1 - x^k) * Kronecker(2, k). (End)
Logarithmic derivative of A111374, the reciprocal of the Goellnitz-Gordon continued fraction: 1+x + x^2/(1+x^3 + x^4/(1+x^5 + x^6/(1+x^7 +...))) in powers of x. - Paul D. Hanna, Jan 10 2014
From Amiram Eldar, Jan 28 2024: (Start)
a(n) = (-1)^(n+1) * A113418(n).
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(24*sqrt(2)) = 0.290786... . (End)

A143259 a(n) = 1 if n is a nonzero square, -1 if n is twice a nonzero square, 0 otherwise.

Original entry on oeis.org

1, -1, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0

Offset: 1

Michael Somos, Aug 02 2008

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

			G.f. = q - q^2 + q^4 - q^8 + q^9 + q^16 - q^18 + q^25 - q^32 + q^36 + q^49 - q^50 + ...

Antti Karttunen, Table of n, a(n) for n = 1..65537
S. Cooper and M. Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 133 Theorem 1.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008836, A010052, A053866, A111374, A154955.

Basis( ModularForms( Gamma1(8), 1/2), 100) [2] ; /* Michael Somos, Jun 10 2014 */

f[n_]:=Which[IntegerQ[Sqrt[n/2]],-1,IntegerQ[Sqrt[n]],1,True,0]; Array[f,110] (* Harvey P. Dale, Jul 07 2011 *)
a[ n_] := Boole[ IntegerQ[ Sqrt[n]]] - Boole[ IntegerQ[ Sqrt[2 n]]]; (* Michael Somos, Jun 10 2014 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2])/2, {q, 0, n}]; (* Michael Somos, Jun 10 2014 *)
Table[LiouvilleLambda[n]*Mod[DivisorSigma[1, n], 2], {n, 100}] (* Jon Maiga, Jan 11 2019 *)

PARI
```
{a(n) = issquare(n) - issquare(2*n)};
```

{a(n) = if( n<1, 0, n--; polcoeff( prod(k=1, n, (1 - x^k)^([1, 1, 0, -1, -1, -1, 0, 1][k%8 + 1]), 1 + x * O(x^n)), n))};

Expansion of (phi(q) - phi(q^2)) / 2 = q * psi(q^4) * f(-q, -q^7) / f(-q^3, -q^5) in powers of q where phi(), psi() and f() are Ramanujan theta functions.
Expansion of q * f(-q, -q^7)^2 / psi(-q) in powers of q where psi(), f() are Ramanujan theta functions. - Michael Somos, Jan 01 2015
Euler transform of period 8 sequence [ -1, 0, 1, 1, 1, 0, -1, -1, ...].
a(2*n) = -a(n) for all n in Z.
a(n) is multiplicative with a(2^e) = (-1)^e, a(p^e) = (1 + (-1)^e) / 2 if p == 1 (mod 2).
Dirichlet g.f.: zeta(2*s) * (1 - 2^-s); Dirichlet convolution of A010052 and A154955.
G.f. A(x) satisfies: A(x) / A(x^2) = -1 + A111374(x).
G.f. A(x) satisfies: A(x^2) = - (A(x) + A(-x)) / 2.
G.f. A(x) satisfies: 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w * (u + v)^2 - v * (v + w) * (v + 2*w).
G.f.: (theta_3(q) - theta_3(q^2)) / 2 = Sum_{k>0} x^(k^2) - x^(2k^2).
|a(n)| = A053866(n).
a(n) = A008836(n)*A053866(n). - Jon Maiga, Jan 11 2019
Sum_{k=1..n} a(k) ~ (1 - 1/sqrt(2)) * sqrt(n). - Vaclav Kotesovec, Oct 16 2020

A245433 Expansion of f(-x^3, -x^5)^2 / (psi(-x) * psi(x^2)) in powers of x where psi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta functions.

Original entry on oeis.org

1, 1, 0, -1, 1, 1, -2, -2, 3, 4, -4, -5, 5, 6, -8, -9, 12, 13, -14, -17, 18, 21, -26, -28, 34, 39, -42, -49, 53, 60, -70, -78, 90, 101, -110, -125, 137, 153, -174, -192, 217, 241, -264, -295, 322, 357, -400, -438, 490, 540, -588, -652, 711, 781, -866, -946

Offset: 0

Michael Somos, Jul 21 2014

sign

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

			G.f. = 1 + x - x^3 + x^4 + x^5 - 2*x^6 - 2*x^7 + 3*x^8 + 4*x^9 + ...
G.f. = 1/q + q^3 - q^11 + q^15 + q^19 - 2*q^23 - 2*q^27 + 3*q^31 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A111374, A245436.

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 1, 1, -2, 1, 1, -1, 0}[[Mod[k, 8, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, Jun 27 2017 *)
f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; a:= CoefficientList[Series[f[-x^3, -x^5]^2/(f[-x, -x^3]*f[x^2, x^6]), {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Aug 06 2018 *)

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[0, -1, 1, 1, -2, 1, 1, -1][k%8 + 1]), n))};

Euler transform of period 8 sequence [1, -1, -1, 2, -1, -1, 1, 0, ...].
Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u^2 - v)^3 - 4 * u^2 * v^3 * (2*v - u^2) * (u^2*v - v^2 - 2).
a(n) = A111374(2*n) = A245436(4*n - 1).

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A327851 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A111374.

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A092869 Series expansion of the Ramanujan-Goellnitz-Gordon continued fraction.

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A069910 Expansion of Product_{i in A069908} 1/(1 - x^i).

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A117000 a(n) = Sum_{d|n} Jacobi(2,d)*d.

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A143259 a(n) = 1 if n is a nonzero square, -1 if n is twice a nonzero square, 0 otherwise.

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A245433 Expansion of f(-x^3, -x^5)^2 / (psi(-x) * psi(x^2)) in powers of x where psi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta functions.

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A111375 Let qf(a,q) = Product(1-aq^j,j=0..infinity); g.f. is qf(q,q^7)qf(q^2,q^7)qf(q^4,q^7)/(qf(q^3,q^7)qf(q^5,q^7)*qf(q^6,q^7)).

A292801 Expansion of 1/(1 + x^2 + x^3/(1 + x^5 + x^7/(1 + x^11 + x^13/(1 + ... + x^prime(2k)/(1 + x^prime(2k+1) + ...))))), a continued fraction.