A122859 -id:A122859 - cp's OEIS frontend

A205972 a(n) = Fibonacci(n)*A122859(n) for n>=1, with a(0)=1, where A122859 lists the coefficients in phi(-q)^3/phi(-q^3) and phi() is a Ramanujan theta function.

1, -6, 12, -12, -18, 0, 96, -156, 252, -204, 0, 0, -864, -2796, 9048, 0, -5922, 0, 31008, -50172, 0, -131352, 0, 0, 556416, -450150, 2913432, -1178508, -3813732, 0, 0, -16155228, 26139708, 0, 0, 0, -89582112, -289893804, 938116056, -758951832, 0, 0, 6429943104

Offset: 0

Paul D. Hanna, Feb 04 2012

sign

Compare the g.f. to the Lambert series of A122859:

1 - 6*Sum_{n>=1} Kronecker(n,3)*x^n/(1+x^n).

			G.f.: A(x) = 1 - 6*x + 12*x^2 - 12*x^3 - 18*x^4 + 96*x^6 - 156*x^7 +...
where A(x) = 1 - 1*6*x + 1*12*x^2 - 2*6*x^3 - 3*6*x^4 + 8*12*x^6 - 13*12*x^7 + 21*12*x^8 - 34*6*x^9 +...+ Fibonacci(n)*A122859(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 - 6*( 1*x/(1+x-x^2) - 1*x^2/(1+3*x^2+x^4) + 3*x^4/(1+7*x^4+x^8) - 5*x^5/(1+11*x^5-x^10) + 13*x^7/(1+29*x^7-x^14) - 21*x^8/(1+47*x^8+x^16) +...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A122859, A205971, A205973, A205966, A205969, A203847, A000204 (Lucas).

Cf. A209452 (Pell variant).

A122859:= CoefficientList[Series[EllipticTheta[3, 0, -q]^3/EllipticTheta[3, 0, -q^3], {q, 0, 60}], q]; Table[If[n == 1, 1, Fibonacci[n - 1]*A122859[[n]]], {n, 1, 50}] (* G. C. Greubel, Dec 03 2017 *)

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 - 6*sum(m=1,n,fibonacci(m)*kronecker(m,3)*x^m/(1+Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,40,print1(a(n),", "))

G.f.: 1 - 6*Sum_{n>=1} Fibonacci(n)*Kronecker(n,3)*x^n/(1 + Lucas(n)*x^n + (-1)^n*x^(2*n)).

A209452 a(n) = Pell(n)*A122859(n) for n>=1, with a(0)=1, where A122859 lists the coefficients in phi(-q)^3/phi(-q^3) and phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -6, 24, -30, -72, 0, 840, -2028, 4896, -5910, 0, 0, -83160, -401532, 1938768, 0, -2824992, 0, 32930520, -79501308, 0, -463367580, 0, 0, 6520076640, -7870428726, 76003583088, -45872220270, -221490672624, 0, 0, -3116610274188, 7524162792576, 0, 0, 0, -127800022137480

Offset: 0

Paul D. Hanna, Mar 10 2012

sign

Compare the g.f. to the Lambert series of A122859: 1 - 6*Sum_{n>=1} Kronecker(n,3)*x^n/(1+x^n).

			G.f.: A(x) = 1 - 6*x + 24*x^2 - 30*x^3 - 72*x^4 + 840*x^6 - 2028*x^7 + ...
where A(x) = 1 - 1*6*x + 2*12*x^2 - 5*6*x^3 - 12*6*x^4 + 70*12*x^6 - 169*12*x^7 + 408*12*x^8 - 985*6*x^9 + ... + Pell(n)*A122859(n)*x^n + ...
The g.f. is also given by the identity:
A(x) = 1 - 6*( 1*x/(1+2*x-x^2) - 2*x^2/(1+6*x^2+x^4) + 12*x^4/(1+34*x^4+x^8) - 29*x^5/(1+82*x^5-x^10) + 169*x^7/(1+478*x^7-x^14) - 408*x^8/(1+1154*x^8+x^16) + ...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A122859, A205972, A209451, A209453, A209446, A209449, A204270, A000129 (Pell), A002203.

A122859[n_]:= SeriesCoefficient[EllipticTheta[4, 0, q]^3/EllipticTheta[4, 0, q^3], {q, 0, n}]; Join[{1}, Table[Fibonacci[n, 2]*A122859[n], {n, 1, 50}]] (* G. C. Greubel, Jan 02 2017 *)

{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
{a(n)=polcoeff(1 - 6*sum(m=1,n,Pell(m)*kronecker(m,3)*x^m/(1+A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,40,print1(a(n),", "))

G.f.: 1 - 6*Sum_{n>=1} Pell(n)*Kronecker(n,3)*x^n/(1 + A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

A093829 Expansion of q * psi(q^3)^3 / psi(q) in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Apr 17 2004

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = q - q^2 + q^3 + q^4 - q^6 + 2*q^7 - q^8 + q^9 + q^12 + 2*q^13 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
James A. Sellers and Roberto Tauraso, Arithmetic properties of MacMahon-type sums of divisors: the odd case, arXiv:2505.12813 [math.NT], 2025. See p. 3.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A033762, A035178, A112300, A113447, A122859.
Cf. A097195, A112604, A112605, A112606, A112607, A112608. - Michael Somos, Sep 27 2013
Cf. A000700, A000122, A010054, A121373.

Basis( ModularForms( Gamma1(6), 1), 90) [2]; /* Michael Somos, Jul 02 2014 */

a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -2, 0, 2, -1, 0} [[ Mod[#, 6, 1]]] &]];
QP = QPochhammer; s = (QP[q]*QP[q^6]^6)/(QP[q^2]^2*QP[q^3]^3) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)

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

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^6 / (eta(x^2 + A)^2 * eta(x^3 + A)^3), n))};

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -12, d) - if( d%2==0, 2 * kronecker( -3, d/2) ) ))}; /* Michael Somos, May 29 2005 */

ModularForms( Gamma1(6), 1, prec=90).1; # Michael Somos, Sep 27 2013

Expansion of (a(q) - a(q^2)) / 6 = c(q^2)^2 / (3 * c(q)) in powers of q where a(), c() are cubic AGM functions. - Michael Somos, Sep 06 2007
Expansion of (eta(q) * eta(q^6)^6) / (eta(q^2)^2 * eta(q^3)^3) in powers of q.
Euler transform of period 6 sequence [ -1, 1, 2, 1, -1, -2, ...].
Moebius transform is period 6 sequence [ 1, -2, 0, 2, -1, 0, ...] = A112300. - Michael Somos, Jul 16 2006
Multiplicative with a(p^e) = (-1)^e if p=2; a(p^e) = 1 if p=3; a(p^e) = 1+e if p == 1 (mod 6); a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 12^(-1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A122859.
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 + 4*w).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u2 * (u2 - u3 - 4*u6) - (u3 + u6) * (u1 - 3*u3 - 3*u6).
G.f.: Sum_{k>0} (x^k - 2 * x^(2*k) + 2 * x^(4*k) - x^(5*k)) / (1 - x^(6*k)) = x * Product_{k>0} ((1 - x^k) * (1 - x^(6*k))^6) / ((1 - x^(2*k))^2 * (1 - x^(3*k))^3).
a(n) = -(-1)^n * A113447(n). - Michael Somos, Jan 31 2015
a(2*n) = -a(n). a(3*n) = a(n). a(6*n + 5) = 0.
A035178(n) = |a(n)|. A033762(n) = a(2*n + 1). A033687(n) = a(3*n + 1).
a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n). a(6*n + 1) = A097195(n). a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(6*sqrt(3)) = 0.302299894039... . - Amiram Eldar, Nov 21 2023

A229616 Expansion of (phi(-q)^3 / phi(-q^3))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -12, 60, -156, 204, -72, -84, -96, 492, -588, 360, -144, 60, -168, 480, -936, 1068, -216, -516, -240, 1224, -1248, 720, -288, 348, -372, 840, -1884, 1632, -360, -504, -384, 2220, -1872, 1080, -576, -372, -456, 1200, -2184, 2952, -504, -672, -528, 2448

Offset: 0

Michael Somos, Sep 26 2013

sign

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 12*q + 60*q^2 - 156*q^3 + 204*q^4 - 72*q^5 - 84*q^6 - 96*q^7 + ...

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
Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.

Cf. A122859, A229615.

A := Basis( ModularForms( Gamma0(6), 2), 50); A[1] - 12*A[2] + 60*A[3];

a[ n_] := If[n < 1, Boole[ n == 0], -12 Sum[ {1, -7, 10, -7, 1, 2}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]];
a[ n_] := If[n < 1, Boole[ n == 0], -12 Sum[ {1, -3, 4, -3, 1, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^6 / EllipticTheta[ 4, 0, q^3]^2, {q, 0, n}];

{a(n) = if( n<1, n==0, -12 * sumdiv( n, d, n/d * [2, 1, -7, 10, -7, 1][d%6 + 1]))};

{a(n) = if( n<1, n==0, -12 * sumdiv( n, d, d * [0, 1, -3, 4, -3, 1][d%6 + 1]))};

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

A = ModularForms( Gamma0(6), 2, prec=50).basis(); A[0] - 12*A[1] + 60*A[2];

Expansion of (2*a(q^2) - a(q))^2 = b(q)^4 / b(q^2)^2 in powers of q where a(), b() are cubic AGM theta functions.
Expansion of (eta(q)^6 * eta(q^6) / (eta(q^2)^3 * eta(q^3)^2))^2 in powers of q.
Euler transform of period 6 sequence [-12, -6, -8, -6, -12, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 432 (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A229615.
G.f.: ( Product_{k>0} (1 + x^(3*k)) * (1 - x^k)^3 / ((1 + x^k)^3 * (1 - x^(3*k))))^2.
Convolution square of A122859.
Conjecture: -3 A122858(n) - A229616(n) + 4 A282031(n) = 0 for all n. - Thomas Baruchel, Jun 23 2018

A122860 Expansion of (1 - phi(-q)^3 / phi(-q^3)) / 6 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Sep 15 2006

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = q - 2*q^2 + q^3 + q^4 - 2*q^6 + 2*q^7 - 2*q^8 + q^9 + q^12 + 2*q^13 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.64).

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.

Cf. A113661, A122859.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

a[ n_] := If[ n < 1, 0, -DivisorSum[ n, (-1)^(n/#) JacobiSymbol[ -3, #] &]]; (* Michael Somos, Feb 19 2015 *)

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

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

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^6 * eta(x^6 + A) / (eta(x^2 + A)^3 * eta(x^3 + A)^2)) / 6, n))};

{a(n) = if( n<1, 0, direuler(p=2, n, if(p==2, (1 - 2*X) / (1 - X^2), 1 / ((1-X) * (1 - kronecker(-3, p)*X))))[n])};

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, (3*(-1)^e - 1) / 2, p==3, 1, p%6==1, e+1, 1-e%2 )))};

Expansion of (1 + a(q) - 2*a(q^2)) / 6 = (1 - b(q)^2 / b(q^2)) / 6 in powers of q where a(), b() are cubic AGM theta functions.
Expansion of (1 - eta(q)^6 * eta(q^6) / (eta(q^2)^3 * eta(q^3)^2)) / 6 in powers of q.
Moebius transform is period 6 sequence [ 1, -3, 0, 3, -1, 0, ...].
a(n) is multiplicative and a(2^e) = (3(-1)^e-1)/2, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
a(3*n) = a(4*n) = a(n). a(6*n + 5) = 0.
G.f.: (1 - Product_{k>0} (1 + x^(3k)) / (1 + x^k)^3 * (1 - x^k)^3 / (1 - x^(3*k))) / 6 = Sum_{k>0} -(-x)^k / (1 + x^k + x^(2*k)).
G.f.: Sum_{k>0} x^(3*k-2) / (1 + x^(3*k-2)) - x^(3*k-1) / (1 + x^(3*k-1)).
-6 * a(n) = A122859(n) unless n=0. -(-1)^n * a(n) = A113661(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 0. - Amiram Eldar, Nov 23 2023

A227354 Expansion of 2 * a(q) - a(q^2) in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

1, 12, -6, 12, 12, 0, -6, 24, -6, 12, 0, 0, 12, 24, -12, 0, 12, 0, -6, 24, 0, 24, 0, 0, -6, 12, -12, 12, 24, 0, 0, 24, -6, 0, 0, 0, 12, 24, -12, 24, 0, 0, -12, 24, 0, 0, 0, 0, 12, 36, -6, 0, 24, 0, -6, 0, -12, 24, 0, 0, 0, 24, -12, 24, 12, 0, 0, 24, 0, 0, 0

Offset: 0

Michael Somos, Jul 08 2013

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 + 12*q - 6*q^2 + 12*q^3 + 12*q^4 - 6*q^6 + 24*q^7 - 6*q^8 + 12*q^9 + ...

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

Cf. A033762, A112604, A112605, A122859, A304656.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^2]^3 / EllipticTheta[ 4, 0, q^6] + 3 EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^3], {q, 0, n}];
a[ n_] := If[ n < 1, Boole[n == 0], 6 Sum[ JacobiSymbol[ d, 3] (Mod[ n/d, 2] + 1), {d, Divisors@n}]]; (* Michael Somos, Jan 09 2015 *)

{a(n) = if( n<1, n==0, 12 * sumdiv( n, d, kronecker( d, 3)) - 6 * sumdiv( 2*n, d, kronecker( d, 3)))};

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 12 * prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, (1 + 3*(-1)^e) / 4, if( p == 3, 1, if( p%6 == 1, e+1, (1 + (-1)^e) / 2 ))))))};

Expansion of (4 * b(q^4)^2 - 2 * b(q) * b(q^4) - b(q)^2) / b(q^2) in powers of q where b() is a cubic AGM theta function.
Expansion of phi(-q^2)^3 / phi(-q^6) + 12 * q * psi(q^2) * psi(q^6) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jan 09 2015
Expansion of theta_4(q^2)^3 / theta_4(q^6) + 3 * theta_2(q) * theta_2(q^3) in powers of q.
Moebius transform is period 6 sequence [ 12, -18, 0, 18, -12, 0, ...].
a(n) = 12 * b(n) where b(n) is multiplicative with b(2^e) = (1 + 3*(-1)^e) / 4, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
a(n) = A122859(8*n). a(2*n) = A122859(n). a(2*n + 1) = 12 * A033762(n). a(4*n) = a(n). a(4*n + 1) = 12 * A112604(n). a(4*n + 2) = -6 * A033762(n). a(4*n + 3) = 12 * A112605(n).
G.f.: 1 + 6 * Sum_{k>0} ((k mod 2) + 1) * x^k / (1 + x^k + x^(2*k)). - Michael Somos, Jan 09 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi*sqrt(3) = 5.441398... (A304656). - Amiram Eldar, Nov 23 2023

A259662 Expansion of phi(-q^3) / phi(-q)^3 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 6, 24, 78, 222, 576, 1392, 3180, 6936, 14550, 29520, 58176, 111750, 209820, 385968, 696960, 1237470, 2163456, 3728904, 6343068, 10658880, 17708412, 29108880, 47373696, 76378992, 122058870, 193435248, 304134558, 474609180, 735374016, 1131698448, 1730375436

Offset: 0

Michael Somos, Jul 02 2015

nonn

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 + 6*x + 24*x^2 + 78*x^3 + 222*x^4 + 576*x^5 + 1392*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A122859, A132974, A132979, A258093.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] / EllipticTheta[ 4, 0, x]^3, {x, 0, n}];
nmax=60; CoefficientList[Series[Product[(1+x^k)^3 * (1-x^(3*k)) / ((1+x^(3*k)) * (1-x^k)^3),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 14 2015 *)

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

Expansion of 1 / (2*a(q^2) - a(q)) = b(q^2) / b(q)^2 in powers of q where a(), b() are cubic AGM theta functions.
Expansion of eta(q^2)^3 * eta(q^3)^2 / (eta(q)^6 * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ 6, 3, 4, 3, 6, 2, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w^2*(u + v)^2 - 2*u*v^2*(v+w).
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 432^(-1/2) (t/I)^-1 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A258093.
G.f.: Product_{k>0} (1 + x^k)^3 * (1 - x^(3*k)) / ((1 + x^(3*k)) * (1 - x^k)^3).
a(n) = A132974(2*n) = A132979(2*n).
Convolution inverse of A122859.
a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(5/4) * n^(5/4)). - Vaclav Kotesovec, Oct 14 2015

cp's OEIS Frontend

A205972 a(n) = Fibonacci(n)*A122859(n) for n>=1, with a(0)=1, where A122859 lists the coefficients in phi(-q)^3/phi(-q^3) and phi() is a Ramanujan theta function.

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A209452 a(n) = Pell(n)*A122859(n) for n>=1, with a(0)=1, where A122859 lists the coefficients in phi(-q)^3/phi(-q^3) and phi() is a Ramanujan theta function.

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A093829 Expansion of q * psi(q^3)^3 / psi(q) in powers of q where psi() is a Ramanujan theta function.

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A229616 Expansion of (phi(-q)^3 / phi(-q^3))^2 in powers of q where phi() is a Ramanujan theta function.

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A122860 Expansion of (1 - phi(-q)^3 / phi(-q^3)) / 6 in powers of q where phi() is a Ramanujan theta function.

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A227354 Expansion of 2 * a(q) - a(q^2) in powers of q where a() is a cubic AGM theta function.

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