A092869 -id:A092869 - cp's OEIS frontend

A079006 Expansion of q^(-1/4) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^2 in powers of q.

1, -2, 5, -10, 18, -32, 55, -90, 144, -226, 346, -522, 777, -1138, 1648, -2362, 3348, -4704, 6554, -9056, 12425, -16932, 22922, -30848, 41282, -54946, 72768, -95914, 125842, -164402, 213901, -277204, 357904, -460448, 590330, -754368, 960948, -1220370

Offset: 0

Michael Somos, Dec 22 2002

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

The Lagrange series reversion of Sum_{n >= 1} a(n-1)*x^n is Sum_{n >= 1} A002103(n-1)*x^n. See the example in A002103. - Wolfdieter Lang, Jul 09 2016

			G.f. A(x) = 1 - 2*x + 5*x^2 - 10*x^3 + 18*x^4 - 32*x^5 + 55*x^6 - 90*x^7 + 144*x^8 + ...
G.f. B(q) = q * A(q^4) = q - 2*q^5 + 5*q^9 - 10*q^13 + 18*q^17 - 32*q^21 + 55*q^25 - 90*q^29 + ...

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
H. R. P. Ferguson, D. E. Nielsen and G. Cook, A partition formula for the integer coefficients of the theta function nome, Math. Comp., 29 (1975), 851-855.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012, eq. (8), p. 11 with eq. (2), p. 10.
Nicco, Conjectured identity of the product of two theta functions, Math Stackexchange, Sep 09 2015.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000700, A001936, A002103, A008441, A029839, A046897, A083365, A127391, A127392, A189925.

a[ n_] := SeriesCoefficient[ Product[(1 + x^(k + 1)) / (1 + x^k), {k, 1, n, 2}]^2, {x, 0, n}]; (* Michael Somos, Jul 08 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ[ q]}, SeriesCoefficient[ (m / 16 / q)^(1/4), {q, 0, n}]]; (* Michael Somos, Jul 08 2011 *)
QP = QPochhammer; s = (QP[q]*(QP[q^4]^2/QP[q^2]^3))^2 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 23 2015 *)
nmax = 50; CoefficientList[Series[Product[(1+x^(2*k))^4 / (1+x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 04 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^4]^2 / QPochhammer[ -x]^2, {x, 0, n}]; (* Michael Somos, Apr 19 2017 *)

{a(n) = my(N, A); if( n<0, 0, N = (sqrtint(16*n + 1) + 1)\2; A = contfracpnqn( matrix(2, N, i, j, if( i==1, if( j<2, 1 + O(x^(N^2 + N)), (x^(j-1) + x^(3*j - 3))^2), 1 - x^(4*j - 2)))); polcoeff( A[2,1] / A[1,1], 4*n))}; /* Michael Somos, Sep 01 2005 */

{a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m = 1; while( m<=n, m*=2; A = subst(A, x, x^2); A = sqrt(A / (1 + 4 * x*A^2))); polcoeff(A, n))};

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

a(n) = (2/n)*Sum_{k=1..n} (-1)^k*A046897(k)*a(n-k). - Vladeta Jovovic, Dec 24 2002
Expansion of q^(-1/4) * (1/2) * k^(1/2) in powers of q, where k^2 is the parameter and q the Jacobi nome of elliptic functions.
Expansion of (1/(2*q)) * (1 - sqrt(k')) / (1 + sqrt(k')) in powers of q^4, where k'^2 is the complementary parameter and q the Jacobi nome of elliptic functions. See the Fricke reference.
Expansion of psi(x^2) / phi(x) = psi(x)^2 / phi(x)^2 = psi(x^2)^2 / psi(x)^2 = psi(-x)^2 / phi(-x^2)^2 = chi(-x)^2 / chi(-x^2)^4 = 1 / (chi(x)^2 * chi(-x^2)^2) = 1 / (chi(x)^4 * chi(-x)^2) = f(-x^4)^2 / f(x)^2 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [-2, 4, -2, 0, ...].
G.f. A(x) satisfies A(x)^2 = A(x^2) / (1 + 4 * x * A(x^2)^2). - Michael Somos, Mar 19 2004
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 * (1 + 4 * v^2) - v. - Michael Somos, Jul 09 2005
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u3 * (u6 + u2)^2 - u2*u6. - Michael Somos, Jul 09 2005
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k-1)))^2 = (Product_{k>0} (1 - x^(4*k)) / (1 - (-x)^k))^2.
Expansion of continued fraction 1 / (1 - x^2 + (x^1 + x^3)^2 / (1 - x^6 + (x^2 + x^6)^2 / (1 - x^10 + (x^3 + x^9)^2 / ...))) in powers of x^4. - Michael Somos, Sep 01 2005
Given g.f. A(x), then B(q) = 2 * q * A(q^4) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (1 - u^4) * (1 - v^4) - (1 - u*v)^4 . - Michael Somos, Jan 01 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A189925.
Convolution inverse of A029839. Convolution square of A083365. a(n) = (-1)^n * A001936(n).
G.f.: 1/Q(0), where Q(k)= 1 - x^(k+1/2) + (x^((k+1)/4) + x^((3*k+3)/4))^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 02 2013
a(n) ~ (-1)^n * exp(Pi*sqrt(n)) / (2^(7/2)*n^(3/4)). - Vaclav Kotesovec, Jul 04 2016
Given g.f. A(x), and B(x) is the g.f. for A008441, then A(x) = B(x^2) / B(x) and A(x) * A(x^2) * A(x^4) * ... = 1 / B(x). - Michael Somos, Apr 20 2017
Expansion of continued fraction 1 / (1 - x^1 + x^1*(1 + x^1)^2 / (1 - x^3 + x^2*(1 + x^2)^2 / (1 - x^5 + x^3*(1 + x^3)^2 / ...))) in powers of x^2. - Michael Somos, Apr 20 2017
a(n) = A208933(4*n+1) - A215348(4*n+1) (conjectured). - Thomas Baruchel, May 14 2018
A(x^4) = (1/(m*x)) * ( chi(x)^m - chi(-x)^m ) / ( chi(x)^m + chi(-x)^m ) at m = 2, where chi(x)  = Product_{i >= 0} (1 + x^(2*i+1)) is the g.f. of A000700. The formula gives generating functions related to A092869 when m = 1 and A001938 (also A093160) when m = 4. - Peter Bala, Sep 23 2023

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)

A093709 Characteristic function of squares or twice squares.

Original entry on oeis.org

1, 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

Offset: 0

Michael Somos, Apr 11 2004

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Partial sums of a(n) for n >= 1 are A071860(n+1). - Jaroslav Krizek, Oct 18 2009
For n > 0, this is also the number of different triangular polyabolos that can be formed from n congruent isosceles right triangles (illustrated at A245676). - Douglas J. Durian, Sep 10 2017

			G.f. = 1 + q + q^2 + q^4 + q^8 + q^9 + q^16 + q^18 + q^25 + q^32 + q^36 + q^49 + ...

Robert Israel, Table of n, a(n) for n = 0..10000
S. Cooper and M. Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 133 Theorem 1.
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. - From _N. J. A. Sloane_, Feb 23 2009
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for characteristic functions

Cf. A000203, A000593, A001227, A028982, A053866, A092869.

A := Basis( ModularForms( Gamma1(8), 1/2), 104); A[1] + A[2]; /* Michael Somos, Jan 01 2015 */

seq(`if`(issqr(n) or issqr(n/2),1,0), n=0..100); # Robert Israel, Apr 05 2016

Table[Boole[IntegerQ[Sqrt[n]] || IntegerQ[Sqrt[2*n]]], {n, 0, 104}] (* Jean-François Alcover, Dec 05 2013 *)
a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors[ n]] || OddQ [ Length @ Divisors[ 2 n]]]]; (* Michael Somos, Jan 01 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}]; (* Michael Somos, Jan 01 2015 *)

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

Expansion of psi(q^4) * f(-q^3, -q^5) / f(-q, -q^7) in powers of q where psi(), f() are Ramanujan theta functions.
Expansion of f(-q^3, -q^5)^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, ...].
G.f. A(x) satisfies A(x^2) = (A(x) + A(-x)) / 2. a(2*n) = a(n).
Given g.f. A(x), then A(x) / A(x^2) = 1 + x*A092869(x^2).
Given g.f. A(x), then B(x) = A(x^2) / A(x) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 + v - 2(u + u^2)*v + 2*(u*v)^2.
Multiplicative with a(0) = a(2^e) = 1, a(p^e) = 1 if e even, 0 otherwise.
a(n) = A053866(n) unless n=0. Characteristic function of A028982 union 0.
G.f.: (theta_3(q) + theta_3(q^2)) / 2 = 1 + (Sum_{k>0} x^(k^2) + x^(2*k^2)).
Dirichlet g.f.: zeta(2*s) * (1 + 2^-s).
For n>0: a(n) = A010052(n) + A010052(A004526(n))*A059841(n). - Reinhard Zumkeller, Nov 14 2009
a(n) = A000035(A000203(n)) = A000035(A000593(n)) = A000035(A001227(n)), if n>0. - Omar E. Pol, Apr 05 2016
Sum_{k=1..n} a(k) ~ (1 + 1/sqrt(2)) * sqrt(n). - Vaclav Kotesovec, Oct 16 2020

A111374 Series expansion of the reciprocal of the Goellnitz-Gordon continued fraction.

Original entry on oeis.org

1, 1, 1, 0, 0, -1, -1, 0, 1, 2, 1, 0, -2, -3, -2, 0, 3, 4, 4, 0, -4, -6, -5, 0, 5, 9, 6, 0, -8, -12, -9, 0, 12, 16, 13, 0, -14, -22, -17, 0, 18, 29, 21, 0, -26, -38, -28, 0, 34, 50, 39, 0, -42, -64, -49, 0, 53, 82, 60, 0, -70, -105, -78, 0, 90, 132, 101, 0, -110, -166, -125, 0, 137, 208, 153, 0, -174, -258, -192, 0, 217

Offset: 0

N. J. A. Sloane, Nov 09 2005

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 15 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Aug 07 2014
A generator (Hauptmodul) of the function field associated with the intersection of congruence subgroups Gamma(2) and Gamma_1(8). [Yang 2004] - Michael Somos, Aug 07 2014

			G.f. = 1 + x + x^2 - x^5 - x^6 + x^8 + 2*x^9 + x^10 - 2*x^12 - 3*x^13 - 2*x^14 + ...
G.f. = 1/q + q + q^3 - q^9 - q^11 + q^15 + 2*q^17 + q^19 - 2*q^23 - 3*q^25 + ...

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.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.

Cf. A003823, A083365, A092869.

M:=100; qf:=(a,q)->mul(1-a*q^j,j=0..M); t2:=qf(q^3,q^8)*qf(q^5,q^8)/(qf(q,q^8)*qf(q^7,q^8)); series(%,q,M); seriestolist(%);

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

{a(n) = my(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( (A^2 + A2) / (2 * A^2 * A2^2 ), n))}; /* Michael Somos, Mar 08 2012 */

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k + x * O(x^n))^-kronecker( 2, k)), n))}; /* Michael Somos, Jul 08 2012 */

Expansion of 1 + x + x^2/(1 + x^3 + x^4/(1 + x^5 + x^6/(1 + x^7+ ...))) in powers of x.
Let qf(a, q) = Product(1-a*q^j, j=0..infinity); g.f. is qf(q^3, q^8)*qf(q^5, q^8)/(qf(q, q^8)*qf(q^7, q^8)).
Expansion of (phi(x) + phi(x^2)) / (2 * psi(x^4)) = 2 * x * 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 f(-x^3, -x^5) / f(-x, -x^7) in powers of x where f(,) is Ramanujan's two-variable theta function. - Michael Somos, Mar 08 2012
Euler transform of period 8 sequence [ 1, 0, -1, 0, -1, 0, 1, 0, ...]. - Michael Somos, Mar 08 2012
Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 * (v - 1) - v * (v + 1). - Michael Somos, Oct 22 2013
a(4*n + 3) = 0. a(4*n + 1) = A083365(n). Convolution inverse of A092869.

A091188 G.f. A(x) satisfies both A(-x)A(x) = A(x^2) and xA(x)^2 = B(xA(x^2)) where B(x) = x(1+x)/(1-x).

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 10, 12, 23, 31, 58, 79, 145, 207, 374, 540, 964, 1427, 2522, 3775, 6626, 10050, 17532, 26811, 46561, 71795, 124188, 192661, 332228, 518303, 891340, 1396902, 2396912, 3771822, 6459202, 10199912, 17437727, 27622807, 47152952

Offset: 0

Paul D. Hanna, Feb 22 2004

nonn

This is a special case of sequences with g.f.s that satisfy the more general functional equation xA(x)^m = B(xA(x^m)) originated by Michael Somos; some other examples are A085748, A091190 and A091200.

			1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 10*x^7 + 12*x^8 + 23*x^9 + ...
q + q^3 + q^5 + 2*q^7 + 2*q^9 + 4*q^11 + 5*q^13 + 10*q^15 + 12*q^17 + ...

Cf. A085748, A091190, A091200, A092869.

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

Given g.f. A(x), then B(x) = x * A(x^2) satisfies 0 = f(B(x), B(x^2)) were f(u, v) = u^2 * (1 - v) - v * (1 + v). - Michael Somos, Aug 02 2011

A226559 Expansion of f(-x^1, -x^7) * f(-x^2, -x^6) / (f(-x^3, -x^5) * f(-x^4, -x^4)) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, -1, -1, 2, 1, -2, -2, 2, 4, -4, -5, 6, 6, -7, -9, 10, 13, -15, -17, 20, 21, -25, -28, 32, 39, -43, -49, 56, 60, -69, -78, 86, 101, -112, -125, 142, 153, -172, -192, 212, 241, -266, -295, 328, 357, -397, -438, 482, 540, -592, -652, 720, 781, -862, -946

Offset: 0

Michael Somos, Jun 10 2013

sign

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

			G.f. = 1 - x - x^2 + 2*x^3 + x^4 - 2*x^5 - 2*x^6 + 2*x^7 + 4*x^8 - 4*x^9 + ...
G.f. = q^3 - q^7 - q^11 + 2*q^15 + q^19 - 2*q^23 - 2*q^27 + 2*q^31 + 4*q^35 + ...

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

Cf. A092869, A230534.

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{0, 1, 1, -1, -2, -1, 1, 1}[[ Mod[k, 8] + 1]], {k, n}], {x, 0, n}];

{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))};

Expansion of (f(-x^1, -x^7) / f(-x^3, -x^5)) * (psi(-x^2) / phi(-x^4)) in powers of x where psi(), phi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [-1, -1, 1, 2, 1, -1, -1, 0, ...].
Given g.f. A(x) then B(q) = q^3 * A(q^4) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (v - u^2)^3 - 4 * u^2 * v^3 * (2*v - u^2) * (2 + v^2 - u^2*v).
a(n) = -A092869(2*n + 1) = A230534(2*n + 1).

A230534 Expansion of phi(x) / phi(x^2) * f(-x, -x^7) / f(-x^3, -x^5) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, -4, -1, 11, -1, -20, 2, 34, 1, -64, -2, 113, -2, -180, 2, 284, 4, -452, -4, 698, -5, -1044, 6, 1545, 6, -2276, -7, 3308, -9, -4724, 10, 6680, 13, -9408, -15, 13130, -17, -18112, 20, 24821, 21, -33864, -25, 45882, -28, -61696, 32, 82514, 39, -109892, -43

Offset: 0

Michael Somos, Oct 22 2013

sign

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

			G.f. = 1 + x - 4*x^2 - x^3 + 11*x^4 - x^5 - 20*x^6 + 2*x^7 + 34*x^8 + ...
G.f. = q + q^3 - 4*q^5 - q^7 + 11*q^9 - q^11 - 20*q^13 + 2*q^15 + ...

Robert G. Wilson v, Table of n, a(n) for n = 0..10000 (first 2501 terms from G. C. Greubel)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A080015, A092869, A226559.

a[ n_] := SeriesCoefficient[ 2 q^(-1/2) EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q]^2 (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2]), {q, 0, n}];

{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)^6) * sum( k=1, sqrtint(n+1), x^k^2 - x^(2*k^2), x * A) / x, n))};

Expansion of (phi(x) / psi(x^2)^2) * (phi(x) - phi(x^2)) / (2*x) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of q^(-1) * phi(q^2) * (phi(q^2) - phi(q^4)) / (phi(q^4) * (phi(q) - phi(q^4))) in powers of q^2 where phi() is a Ramanujan theta function.
Euler transform of period 8 sequence [1, -5, 3, 2, 3, -5, 1, 0, ...].
a(2*n + 1) = A226559(n).
Convolution of A080015 and A092869.

A230535 Expansion of q * (f(-q, -q^7) / f(-q^3, -q^5))^2 in powers of q where f(,) is Ramanujan's two-variable theta function.

Original entry on oeis.org

1, -2, 1, 2, -4, 4, -1, -6, 11, -8, -1, 12, -20, 16, 2, -22, 34, -30, 1, 40, -64, 52, -2, -68, 113, -88, -2, 112, -180, 144, 2, -182, 284, -228, 4, 286, -452, 356, -4, -440, 698, -544, -5, 668, -1044, 816, 6, -996, 1545, -1210, 6, 1464, -2276, 1768, -7, -2128

Offset: 1

Michael Somos, Oct 22 2013

sign

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

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

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. A092869, A224216, A230534.

a[ n_] := SeriesCoefficient[ q Product[ (1 - q^k)^KroneckerSymbol[ 8, k], {k, n}]^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^2] / (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2]))^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q, q^8] QPochhammer[ q^7, q^8] / (QPochhammer[ q^3, q^8] QPochhammer[ q^5, q^8]))^2, {q, 0, n}];

{a(n) = local(A, A2); if( n<2, n==1, n--; 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))^2, n))};

Euler transform of period 8 sequence [ -2, 0, 2, 0, 2, 0, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v - u^2) * (1 - v)  - 4*u*v.
G.f.: x * (Product_{k>=0} (1 - x^(8*k + 1)) * (1 - x^(8*k + 7)) / ((1 - x^(8*k + 3)) * (1 - x^(8*k + 5))))^2.
a(2*n) = -2 * A224216(n). a(2*n + 1) = A230534(n).
Convolution square of A092869.
Empirical: Sum_{n>=1} a(n)/exp(Pi*n) = 7 + 4*sqrt(2) - 2*sqrt(20 + 14*sqrt(2)). - Simon Plouffe, Mar 02 2021

cp's OEIS Frontend

A327852 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A092869.

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A079006 Expansion of q^(-1/4) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^2 in powers of q.

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

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A093709 Characteristic function of squares or twice squares.

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A111374 Series expansion of the reciprocal of the Goellnitz-Gordon continued fraction.

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A091188 G.f. A(x) satisfies both A(-x)*A(x) = A(x^2) and xA(x)^2 = B(xA(x^2)) where B(x) = x*(1+x)/(1-x).

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A226559 Expansion of f(-x^1, -x^7) * f(-x^2, -x^6) / (f(-x^3, -x^5) * f(-x^4, -x^4)) in powers of x where f(, ) is Ramanujan's general theta function.

A091188 G.f. A(x) satisfies both A(-x)A(x) = A(x^2) and xA(x)^2 = B(xA(x^2)) where B(x) = x(1+x)/(1-x).

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.