A113973 -id:A113973 - cp's OEIS frontend

A205969 a(n) = Fibonacci(n)*A113973(n) for n>=1, with a(0)=1, where A113973 lists the coefficients in phi(x^3)^3/phi(x) and phi() is a Ramanujan theta function.

1, -2, 4, -4, 6, 0, 32, -52, 84, -68, 0, 0, 288, -932, 3016, 0, 1974, 0, 10336, -16724, 0, -43784, 0, 0, 185472, -150050, 971144, -392836, 1271244, 0, 0, -5385076, 8713236, 0, 0, 0, 29860704, -96631268, 312705352, -252983944, 0, 0, 2143314368, -1733977748, 0

Offset: 0

Paul D. Hanna, Feb 04 2012

sign

Compare g.f. to the Lambert series of A113973: 1 - 2*Sum_{n>=1} Kronecker(n,3)*x^n/(1 - (-x)^n).

			G.f.: A(x) = 1 - 2*x + 4*x^2 - 4*x^3 + 6*x^4 + 32*x^6 - 52*x^7 + 84*x^8 +...
where A(x) = 1 - 1*2*x + 1*4*x^2 - 2*2*x^3 + 3*2*x^4 + 8*4*x^6 - 13*4*x^7 + 21*4*x^8 +...+ Fibonacci(n)*A113973(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 - 2*( 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..2500

Cf. A113973, A205966, A205968, A205970, A203847, A000204 (Lucas).

Cf. A209449 (Pell variant).

A113973:= CoefficientList[Series[EllipticTheta[3, q^3]^3/EllipticTheta[3, 0, q], {q, 0, 75}], q]; Table[If[n == 1, 1, Fibonacci[n-1]*A113973[[n]] ], {n, 1, 50}] (* G. C. Greubel, Jul 17 2018 *)

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 - 2*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=0,60,print1(a(n),", "))

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

A209449 a(n) = Pell(n)*A113973(n) for n>=1, with a(0)=1, where A113973 lists the coefficients in phi(x^3)^3/phi(x) and phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 8, -10, 24, 0, 280, -676, 1632, -1970, 0, 0, 27720, -133844, 646256, 0, 941664, 0, 10976840, -26500436, 0, -154455860, 0, 0, 2173358880, -2623476242, 25334527696, -15290740090, 73830224208, 0, 0, -1038870091396, 2508054264192, 0, 0, 0, 42600007379160

Offset: 0

Paul D. Hanna, Mar 10 2012

sign

Compare g.f. to the Lambert series of A113973:

1 - 2*Sum_{n>=1} Kronecker(n,3)*x^n/(1 - (-x)^n).

			G.f.: A(x) = 1 - 2*x + 8*x^2 - 10*x^3 + 24*x^4 + 280*x^6 - 676*x^7 +...
where A(x) = 1 - 1*2*x + 2*4*x^2 - 5*2*x^3 + 12*2*x^4 + 70*4*x^6 - 169*4*x^7 + 408*4*x^8 +...+ Pell(n)*A113973(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 - 2*( 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. A113973, A205969, A209446, A209448, A209450, A204270, A000129 (Pell), A002203.

A113973:= CoefficientList[Series[EllipticTheta[3, 0, q^3]^3 /EllipticTheta[3, 0, q], {q, 0, 60}], q]; Table[If[n == 0, 1, Fibonacci[n, 2]*A113973[[n + 1]]], {n, 0, 50}] (* G. C. Greubel, Dec 03 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 - 2*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=0,60,print1(a(n),", "))

G.f.: 1 - 2*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).

A164617 Expansion of (phi^3(q^3) / phi(q)) * (psi(-q^3) / psi^3(-q)) in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 4, 10, 20, 39, 76, 140, 244, 415, 696, 1140, 1820, 2861, 4448, 6816, 10292, 15372, 22756, 33356, 48408, 69683, 99600, 141312, 199036, 278557, 387608, 536230, 737632, 1009464, 1374888, 1863764, 2514868, 3378948, 4521672, 6027000, 8002676

Offset: 0

Michael Somos, Aug 17 2009

nonn

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

			G.f. = 1 + q + 4*q^2 + 10*q^3 + 20*q^4 + 39*q^5 + 76*q^6 + 140*q^7 + 244*q^8 + ...

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. A113973, A128641, A132974, A164616.

nmax=60; CoefficientList[Series[Product[(1-x^(6*k))^14 / ((1-x^k) * (1-x^(2*k))^2 * (1-x^(3*k))^5 * (1-x^(4*k)) * (1-x^(12*k))^5),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^14 / (eta(x + A) * eta(x^2 + A)^2 * eta(x^3 + A)^5 * eta(x^4 + A) * eta(x^12 + A)^5), n))};

Expansion of eta(q^6)^14 / (eta(q) * eta(q^2)^2 * eta(q^3)^5 * eta(q^4) * eta(q^12)^5) in powers of q.
Euler transform of period 12 sequence [ 1, 3, 6, 4, 1, -6, 1, 4, 6, 3, 1, 0, ...].
Convolution of A113973 and A132974. a(n) = A164616(3*n).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
A128641(n) = (-1)^n*a(n). - Michael Somos, Apr 24 2023

A123330 Expansion of eta(q^2) * eta(q^3)^6 / (eta(q)^2 * eta(q^6)^3) in powers of q.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Sep 26 2006

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

			G.f. = 1 + 2*q + 4*q^2 + 2*q^3 + 2*q^4 + 4*q^6 + 4*q^7 + 4*q^8 + 2*q^9 + ... - _Michael Somos_, Aug 11 2009

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. A033687, A033762, A097195, A107760, A112604, A112605, A113973, A121361, A123331, A123884, A275486.

Cf. A000700, A000122, A010054, A121373.

QP = QPochhammer; s = QP[q^2]*(QP[q^3]^6/(QP[q]^2*QP[q^6]^3)) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

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

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

A = ModularForms( Gamma1(6), 1, prec=90).basis(); A[0] + 2*A[1] # Michael Somos, Sep 27 2013

Expansion of c(q)^2 / (3 * c(q^2)) in powers of q where c() is a cubic AGM theta function.
Expansion of phi(-x^3)^3 / phi(-x) where phi() is a Ramanujan theta function.
a(n) = 2*b(n) where b(n) is multiplicative and b(2^e) = (1 - 3*(-1)^e) / 2 if e>0, 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).
Euler transform of period 6 sequence [ 2, 1, -4, 1, 2, -2, ...].
Moebius transform is period 6 sequence [ 2, 2, 0, -2, -2, 0, ...].
a(n) = 2 * A123331(n) if n>0. (-1)^n * a(n) = A113973(n).
G.f.: Product_{k>0} (1 + x^k)/(1 - x^k) * ((1 - x^(3*k)) / (1 + x^(3*k)))^3.
G.f.: 1 + 2 * Sum_{k>0} x^k / (1 - x^k + x^(2*k)) = theta_3(-x^3)^3 / theta_3(-x).
From Michael Somos, Aug 11 2009: (Start)
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v * (u - v)^2 - 2 * u * w * (v - w).
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (16/3)^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A107760.
a(4*n) = a(3*n) = a(n). a(12*n + 10) = a(6*n + 5) = 0.
a(2*n + 1) = 2 * A033762(n). a(3*n + 1) = 2 * A033687(n). a(4*n + 1) = 2 * A112604(n). a(4*n + 3) = 2 * A112605(n). a(6*n + 1) = 2 * A097195(n). a(12*n + 1) = A123884(n). a(12*n + 7) = 4 * A121361(n). (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/(3*sqrt(3)) = 2.418399... (A275486). - Amiram Eldar, Nov 14 2023

A113660 Expansion of phi(x)^3 / phi(x^3) where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 6, 12, 6, -6, 0, 12, 12, 12, 6, 0, 0, -6, 12, 24, 0, -6, 0, 12, 12, 0, 12, 0, 0, 12, 6, 24, 6, -12, 0, 0, 12, 12, 0, 0, 0, -6, 12, 24, 12, 0, 0, 24, 12, 0, 0, 0, 0, -6, 18, 12, 0, -12, 0, 12, 0, 24, 12, 0, 0, 0, 12, 24, 12, -6, 0, 0, 12, 0, 0, 0, 0, 12, 12

Offset: 0

Michael Somos, Nov 03 2005

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*q + 12*q^2 + 6*q^3 - 6*q^4 + 12*q^6 + 12*q^7 + 12*q^8 + 6*q^9 + ...

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 227, Entry 4(iv).

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. A113661, A113973, A304656.

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

A := Basis( ModularForms( Gamma1(12), 1), 74); A[1] + 6*A[2] + 12*A[3] + 6*A[4] - 6*A[5]; /* Michael Somos, May 20 2015 */

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3 / EllipticTheta[ 3, 0, q^3], {q, 0, n}]; (* Michael Somos, May 20 2015 *)

{a(n) = my(x); if( n<1, n==0, x = valuation(n, 2); if( n%2, 2, (1 - 3*(-1)^x))*3 * sumdiv(n/2^x, d, kronecker(-3, d)))};

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

{a(n) = if( n<1, n==0, 6 * direuler(p=2, n, if( p==2, 2 - (1 - 2*X) / (1 - X^2), 1 / ((1 - X) * (1 - kronecker(-3, p) * X))))[n])};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^15 * eta(x^3 + A)^2 * eta(x^12 + A)^2 / (eta(x + A)^6 * eta(x^4 + A)^6 * eta(x^6 + A)^5), n))};

Expansion of a(q) + 2*q(q^2) - 2*a(q^4) = b(-q)^2 / b(q^2) = (b(q) - 2*b(q^4))^2 / b(q^2) = (c(q) + 2*c(q^4))^2 / (3 * c(q^2)) in powers of q where a(), b(), c() are cubic AGM functions. - Michael Somos, May 20 2015
Expansion of (eta(q^2)^15 * eta(q^3)^2 * eta(q^12)^2) / (eta(q)^6 * eta(q^4)^6 * eta(q^6)^5) in powers of q.
a(n) = 6*b(n) where b(n) is multiplicative with a(0) = 1, b(2^e) = (1 - 3(-1)^e) / 2 if e>0, 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).
Euler transform of period 12 sequence [ 6, -9, 4, -3, 6, -6, 6, -3, 4, -9, 6, -2, ...].
Moebius transform is period 12 sequence [ 6, 6, 0, -18, -6, 0, 6, 18, 0, -6, -6, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113973. - Michael Somos, May 20 2015
G.f.: 1 + 6 * ( Sum_{k>0} x^k / (1 + x^k + x^(2*k)) + 2*x^(4*k - 2) / (1 + x^(4*k - 2) + x^(8*k - 4)) ).
a(n) = 6 * A113661(n), if n>0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi*sqrt(3) = 5.441398... (A304656). - Amiram Eldar, Dec 25 2023

Corrected by Charles R Greathouse IV, Sep 02 2009

A113974 Expansion of (1-phi(x^3)^3/phi(x))/2 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, 0, 0, -2, 0, -4, 0, 0, 0, 0, 4, 0, 2, 0, 0, -2, 2, -6, 0, -1, 0, 0, 2, -4, 0

Offset: 1

Michael Somos, Nov 10 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).

Bruce C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, 1985, see p. 375, Entry 35.

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. A000700, A000122, A010054, A113973, A121373.

a[n_]:= SeriesCoefficient[(1 - EllipticTheta[3, 0, q^3]^3/EllipticTheta[ 3, 0, q])/2, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Dec 16 2017 *)
f[p_, e_] := If[Mod[p, 6] == 1, e + 1, (1 + (-1)^e)/2]; f[2, e_] := ((-1)^e - 3)/2; f[3, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 14 2023 *)

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

{a(n)=local(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, (-3+(-1)^e)/2, if(p==3, 1, if(p%6==1, e+1, !(e%2)))))))}

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

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

a(n) is multiplicative and a(2^e) = ((-1)^e-3)/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). [corrected by Amiram Eldar, Nov 14 2023]
Moebius transform is period 12 sequence [1, -3, 0, 1, -1, 0, 1, -1, 0, 3, -1, 0, ...].
G.f.: (1-theta_3(q^3)^3/theta_3(q))/2.
G.f.: Sum_{k>0} x^(3k-2)/(1-(-x)^(3k-2)) - x^(3k-1)/(1-(-x)^(3k-1)) = Sum_{k>0} -(-1)^k x^k/(1+x^k+x^(2k)) -2 x^(4k)/(1+x^(4k)+x^(8k)).
-2*a(n) = A113973(n), if n>0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = -Pi/(6*sqrt(3)) = -0.302299... . - Amiram Eldar, Nov 14 2023

A226132 Expansion of - c(-q) * c(q^2) / 9 in powers of q where c() is a cubic AGM theta function.

Original entry on oeis.org

1, -1, 3, -1, 6, -3, 8, -1, 9, -6, 12, -3, 14, -8, 18, -1, 18, -9, 20, -6, 24, -12, 24, -3, 31, -14, 27, -8, 30, -18, 32, -1, 36, -18, 48, -9, 38, -20, 42, -6, 42, -24, 44, -12, 54, -24, 48, -3, 57, -31, 54, -14, 54, -27, 72, -8, 60, -30, 60, -18, 62, -32, 72

Offset: 1

Michael Somos, May 27 2013

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

Number 91 of the 126 eta-quotients listed in Table 1 of Williams 2012.

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

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.
K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

Cf. A113447, A113973, A121443, A185717, A226139.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 1, 0, Sum[ If[ OddQ[d] && ! Divisible[ n/d, 3], -d (-1)^(n/d), 0], {d, Divisors[ n]}]];
a[ n_] := If[ n < 2, Boole[n == 1], Times @@ (Which[ # == 2, -1, # == 3, #^#2, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger[n])];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3]^3 / EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, I q^(3/2)]^3 / EllipticTheta[ 2, 0, I q^(1/2)] / (4 (-1)^(1/4)), {q, 0, n}];

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

{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( p==3, p^e, (p^(e+1) - 1) / (p - 1))))))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^12 / (eta(x^2 + A)^4 * eta(x^3 + A)^3 * eta(x^12 + A)^3), n))};

Expansion of (a(q) - a(q^2)) * (a(q^2) + 2 * a(q^4)) / 18 = c(q^2)^4 / (9 * c(q) * c(q^4)) = (b(-q) * b(q^2) - a(-q) * a(q^2)) / 9 in powers of q where a(), b(), c(q) are cubic AGM theta functions.
Expansion of q * (phi(q^3)^3 / phi(q)) * (ps(-q^3)^3 / psi(-q)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^4) * eta(q^6)^12 / (eta(q^2)^4 * eta(q^3)^3 * eta(q^12)^3) in powers of q.
Euler transform of period 12 sequence [ -1, 3, 2, 2, -1, -6, -1, 2, 2, 3, -1, -4, ...].
Multiplicative with a(2^e) = -1 if e>0, a(3^e) = 3^e, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 4/3 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226139.
G.f.: Sum_{k>0 not 3|k} x^k / (1 - (-x)^k)^2 = Sum_{k>0 not 2|k} k * x^k * (1 - x^k) / (1 + x^(3*k)).
G.f.: x * Product_{k>0} (1 - x^k) * (1 - x^(4*k)) * (1 - x^(3*k))^6 * (1 + x^(3*k))^9 / ((1 - x^(2*k))^4 * (1 + x^(6*k))^3).
a(2*n) = - A121443(n). a(2*n + 1) = A185717(n).
a(n) = -(-1)^n * A121443(n). Convolution of A113447 and A113973.
From Amiram Eldar, Sep 12 2023: (Start)
Dirichlet g.f.: (1 - 1/2^(s-1))^2 * (1 - 1/3^s) * zeta(s-1) * zeta(s).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/54 = 0.18277045... . (End)

A119428 G.f.: A(x) = 1 + Sum_{n>=0} (-1)^n* Sum_{k=1..4} x^(5n+k)/(1-x^(5n+k)).

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, May 19 2006

sign

Records are A003586 (numbers of the form 2^i*3^j) = [1,2,3,4,6,8,9,12...], occurring at positions: [0,2,4,12,44,132,484,924,4092,...].

			A(x) = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + x^5 + 2*x^6 + 2*x^8 + x^9 +...
= 1 + x/(1-x) + x^2/(1-x^2) + x^3/(1-x^3) + x^4/(1-x^4) +
- x^6/(1-x^6) - x^7/(1-x^7) - x^8/(1-x^8) - x^9/(1-x^9) +
+ x^11/(1-x^11) + x^12/(1-x^12) + x^13/(1-x^13) + x^14/(1-x^14) +...

G. C. Greubel, Table of n, a(n) for n = 0..1000
L.-C. Shen, On the additive formulas of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5, Trans. Amer. Math. Soc. 345 (1994), no. 1, 323-345. See p. 342, Eq. (3.40).

Cf. A113973.

A119428[n_] := SeriesCoefficient[(QPochhammer[q^5,q^5])^2/(QPochhammer[q, q^10]*QPochhammer[q^2, q^10]*QPochhammer[q^8, q^10]*QPochhammer[q^9, q^10]), {q, 0, n}]; Table[A119428[n], {n, 0, 50}] (* G. C. Greubel, Oct 03 2017 *)

{a(n)=polcoeff(1+sum(k=0,n\5+1,(-1)^k*sum(j=1,4,x^(5*k+j)/(1-x^(5*k+j)+x*O(x^n))) ),n)}

{a(n) = local(A); if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k+ x*O(x^n))^[2,-1,-1,0,0,2,0,0,-1,-1][k%10 + 1]), n))} /* Michael Somos, Mar 08 2008 */

a(5n) = a(n); a(10n+2) = 2*a(5n+1).
Convolution identity: Sum_{k=0..n} a(5(n-k))*a(5k+3) = Sum_{k=0..n} a(5(n-k)+1)*a(5k+2).
Euler transform of period 10 sequence: [1,1,0,0,-2,0,0,1,1,-2]; also,
Moebius transform of period 10 sequence: [1,1,1,1,0,-1,-1,-1,-1,0].
Expansion of (f(-q^5) * f(-q^10))^2 / (f(-q, -q^9) * f(-q^2, -q^8)) in powers of q where f() is Ramanujan's two-variable theta function.  - Michael Somos, Mar 08 2008
G.f.: Product_{k>0} (1 - x^(5*k))^2 / ( (1-x^(10*k-1)) * (1-x^(10*k-2)) * (1-x^(10*k-8)) * (1-x^(10*k-9)) ). - Michael Somos, Mar 08 2008
G.f.: 1 + Sum_{k>0} (x^k + x^(2*k) + x^(3*k) + x^(4*k)) / (1 + x^(5*k)). - Michael Somos, Mar 08 2008
If A = A0 + A1 + A2 + A3 + A4 is the 5-section, then A0 * A3 = A1 * A2, A0 * A2 + A3 * A4 = 2 * A1^2. - Michael Somos, Mar 08 2008

A321466 Expansion of (phi(x^3)^3 / phi(x))^2 in powers of x where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 12, -20, 28, -24, 28, -32, 60, -68, 72, -48, 44, -56, 96, -120, 124, -72, 76, -80, 168, -160, 144, -96, 76, -124, 168, -212, 224, -120, 168, -128, 252, -240, 216, -192, 92, -152, 240, -280, 360, -168, 224, -176, 336, -408, 288, -192, 140, -228, 372

Offset: 0

Michael Somos, Nov 11 2018

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).
The g.f. of A113973 is k = k(q) := phi(x^3)^3 / phi(x) given in equation (2.2) page 996 of Williams 2012, and the g.f. of k^2 which is given in equation (2.3) page 997 is this sequence.
Number 54 of the 126 eta-quotients listed in Table 1 of Williams 2012.

			G.f. = 1 - 4*x + 12*x^2 - 20*x^3 + 28*x^4 - 24*x^5 + 28*x^6 + ...

Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

Cf. A113973,A227226,A321465.

A := Basis( ModularForms( Gamma0(12), 2), 51); A[1] - 4*A[2] + 12*A[3] - 20*A[4] + 28*A[5];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3]^6 / EllipticTheta[ 3, 0, x]^2, {x, 0, n}];
a[ n_] := With[{s = If[ FractionalPart @ # > 0, 0, DivisorSigma[1, #]] &}, If[ n < 1, Boole[n == 0], -4 (s[n] - 6 s[n/2] + s[n/3] + 4 s[n/4] + 2 s[n/6] + 4 s[n/12])]];

{a(n) = my(s = x -> if(frac(x), 0, sigma(x))); if( n<1, n==0, -4*(s(n) - 6*s(n/2) + s(n/3) + 4*s(n/4) + 2*s(n/6) + 4*s(n/12)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^15 / (eta(x^2 + A)^5 * eta(x^3 + A)^6 * eta(x^12 + A)^6))^2, n))};

Expansion of (eta(q)^2 * eta(q^4)^2 * eta(q^6)^15 / (eta(q^2)^5 * eta(q^3)^6 * eta(q^12)^6))^2 in powers of q.
Expansion of ((a(x) - 2*a(x^2) - 2*a(x^4))/3)^2 = ((b(x) + 2*b(x^4))^2 / (9*b(x^2)))^2 in powers of x where a(), b() are cubic AGM theta functions.
Euler transform of period 12 sequence [-4, 6, 8, 2, -4, -12, -4, 2, 8, 6, -4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (4/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A321465.
G.f.: (theta_3(0, x^3)^3 / theta_3(0, x))^2 where theta_3(0, x) is a Jacobi theta function.
G.f.: (Product_{k>0} f(x^k))^2 where f(x) := ((1 - x) * (1 + x^2))^2 * ((1 - x^3) * (1 + x^3)^3)^3 / ((1 - x^2) * (1 + x^6)^2)^3.
a(n) = -4*(s(n) - 6*s(n/2) + s(n/3) + 4*s(n/4) + 2*s(n/6) + 4*s(n/12)) if n>0 where s(x) = sum of divisors of x for integer x else 0.
a(n) = (-1)^n * A227226(n). Convolution square of A113973.

cp's OEIS Frontend

A205969 a(n) = Fibonacci(n)*A113973(n) for n>=1, with a(0)=1, where A113973 lists the coefficients in phi(x^3)^3/phi(x) and phi() is a Ramanujan theta function.

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A209449 a(n) = Pell(n)*A113973(n) for n>=1, with a(0)=1, where A113973 lists the coefficients in phi(x^3)^3/phi(x) and phi() is a Ramanujan theta function.

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A164617 Expansion of (phi^3(q^3) / phi(q)) * (psi(-q^3) / psi^3(-q)) in powers of q where phi(), psi() are Ramanujan theta functions.

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A123330 Expansion of eta(q^2) * eta(q^3)^6 / (eta(q)^2 * eta(q^6)^3) in powers of q.

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A113660 Expansion of phi(x)^3 / phi(x^3) where phi() is a Ramanujan theta function.

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A113974 Expansion of (1-phi(x^3)^3/phi(x))/2 where phi() is a Ramanujan theta function.

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