A111932 -id:A111932 - cp's OEIS frontend

A186100 Expansion of 2 * a(q^2)^2 - a(q)^2 in powers of q where a() is a cubic AGM theta function.

1, -12, -12, -12, -12, -72, -12, -96, -12, -12, -72, -144, -12, -168, -96, -72, -12, -216, -12, -240, -72, -96, -144, -288, -12, -372, -168, -12, -96, -360, -72, -384, -12, -144, -216, -576, -12, -456, -240, -168, -72, -504, -96, -528, -144, -72, -288

Offset: 0

Michael Somos, Feb 12 2011

sign

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008653, A098098, A111932, A121443, A131943, A131946, A186099.

a[ n_] := If[ n < 1, Boole[n == 0], -12 DivisorSum[ n, # Boole[ 1 == GCD[#, 6]] &]]; (* Michael Somos, Jul 07 2015 *)
a[ n_] := SeriesCoefficient[(EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^3])^2 - 1/2 (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)])^2, {x, 0, n}]; (* Michael Somos, Jul 07 2015 *)

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

{a(n) = if( n<1, n==0, -12 * direuler( p=2, n, 1 / (1 - X) / (1 - (p>3) * p * X)) [n])};

Expansion of b(x) * b(x^2) - c(x) * c(x^2) in powers of x where b(), c() are cubic AGM functions.
Expansion of (phi(-x) * phi(-x^3))^2 - 8 * x * (psi(x) * psi(x^3))^2 in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of (P(q) - 2*P(q^2) - 3*P(q^3) + 6*P(q^6)) / 2 in powers of q where P() is a Ramanujan Eisenstein series. - Michael Somos, Jul 07 2015
a(n) = -12 * A186099(n) if n>0. a(2*n) = a(n). a(2*n + 1) = - A008653(2*n + 1). a(n) = 2 * A008653(n) - A008653(2*n) = A131946(n) - 8 * A111932(n) = A131943(n) - 9 * A121443(n).
a(3*n) = a(n). a(6*n + 5) = -72 * A098098(n).- Michael Somos, Jul 07 2015

A131946 Expansion of (phi(-q) * phi(-q^3))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 4, -4, 20, -24, 4, -32, 52, -4, 24, -48, 20, -56, 32, -24, 116, -72, 4, -80, 120, -32, 48, -96, 52, -124, 56, -4, 160, -120, 24, -128, 244, -48, 72, -192, 20, -152, 80, -56, 312, -168, 32, -176, 240, -24, 96, -192, 116, -228, 124, -72, 280, -216, 4

Offset: 0

Michael Somos, Jul 30 2007

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 - 4*q + 4*q^2 - 4*q^3 + 20*q^4 - 24*q^5 + 4*q^6 - 32*q^7 + 52*q^8 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.66).

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. A034896, A111932, A125514, A131947.

A := Basis( ModularForms( Gamma0(6), 2), 55); A[1] - 4*A[2] + 4*A[3]; /* Michael Somos, Nov 11 2015 */

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3])^2, {q, 0, n}]; (* Michael Somos, Sep 19 2013 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^3])^4 / (QPochhammer[ q^2] QPochhammer[ q^6])^2, {q, 0, n}]; (* Michael Somos, Sep 19 2013 *)
a[ n_] := If[ n < 1, Boole[n == 0], -4 Sum[ d {0, 1, -1, 0, -1, 1}[[ Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* Michael Somos, Sep 19 2013 *)
a[ n_] := If[ n < 1, Boole[n == 0], -4 Sum[ n/d {6, 1, -3, -2, -3, 1}[[ Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* Michael Somos, Sep 19 2013 *)

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

{a(n) = if( n<1, n==0, -4 * sumdiv( n, d, d * [0, 1, -1, 0, -1, 1][d%6 + 1]))}; /* Michael Somos, Sep 19 2013 */

A = ModularForms( Gamma0(6), 2, prec=55) . basis();  A[0] - 4*A[1] + 4*A[2]; # Michael Somos, Sep 19 2013

Expansion of (4*a(q^2)^2 - a(q)^2) / 3 in powers of q where a() is a cubic AGM theta function.
Expansion of (b(q)^2 / b(q^2)) * (c(q)^2 / c(q^2)) / 3 in powers of q where b(), c() are cubic AGM theta functions.
Expansion of (eta(q) * eta(q^3))^4 / ( eta(q^2) * eta(q^6))^2 in powers of q.
Euler transform of period 6 sequence [-4, -2, -8, -2, -4, -4, ...].
a(n) = -4 * b(n) where b() is multiplicative with b(2^e) = 3 - 2^(e+1), b(3^e) = 1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3. - Michael Somos, Sep 19 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 48 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A111932. - Michael Somos, Sep 19 2013
G.f.: 1 - 4 * (Sum_{k>0} k * (-x)^k / (1 - x^k) * Kronecker(9, k)) = (theta_3(-x) * theta_3(-x^3))^2.
a(n) = (-1)^n * A034896(n). a(n) = -4 * A131947(n) unless n = 0.
a(3*n) = a(n). a(2*n) = A125514(n). - Michael Somos, Sep 19 2013

A232356 Expansion of 2/9 * c(q) * c(q^2) - q * (psi(q) * psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function and c(q) is a cubic AGM theta function.

Original entry on oeis.org

1, 0, 5, -2, 6, 4, 8, -6, 17, 0, 12, 2, 14, 0, 30, -14, 18, 16, 20, -12, 40, 0, 24, -2, 31, 0, 53, -16, 30, 24, 32, -30, 60, 0, 48, 14, 38, 0, 70, -36, 42, 32, 44, -24, 102, 0, 48, -10, 57, 0, 90, -28, 54, 52, 72, -48, 100, 0, 60, 12, 62, 0, 136, -62, 84, 48

Offset: 1

Michael Somos, Nov 22 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. = q + 5*q^3 - 2*q^4 + 6*q^5 + 4*q^6 + 8*q^7 - 6*q^8 + 17*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008438, A033686, A098098, A111932, A121443, A123532, A144614, A229615, A232343.

Basis( ModularForms( Gamma0(6), 2), 70) [2];

a[ n_] := If[ n < 1, 0, Sum[ d ( 2 Mod[ d, 2] Boole[Mod[ n/d, 3] > 0] - Mod[ n/d, 2] Boole[ Mod[d, 3] > 0]), {d, Divisors @n}]];
a[ n_] := SeriesCoefficient[ 2 q (QPochhammer[ q^3] QPochhammer[ q^6])^3 / (QPochhammer[ q] QPochhammer[ q^2]) - q (QPochhammer[ q^2] QPochhammer[ q^6])^4 / (QPochhammer[ q] QPochhammer[ q^3])^2, {q, 0, n}];

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

ModularForms( Gamma0(6), 2, prec=70).1;

a(n) = 2 * A121443(n) - A111932(n). a(2*n) = -2 * A229615(n). a(12*n + 2) = a(12*n + 10) = 0.

a(n) = A123532(n) + 7 * A229615(n). a(3*n + 2) = 6 * A232343(n-1). a(6*n + 5) = 6 * A098098(n). a(12*n + 4) = -2 * A144614(n). a(12*n + 6) = 4 * A008438(n). a(12*n + 8) = -6 * A033686(n). - Michael Somos, May 23 2014

A121456 Expansion of q(psi(-q)psi(-q^3))^2 in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 1, -4, 6, -2, 8, -8, 1, -12, 12, -4, 14, -16, 6, -16, 18, -2, 20, -24, 8, -24, 24, -8, 31, -28, 1, -32, 30, -12, 32, -32, 12, -36, 48, -4, 38, -40, 14, -48, 42, -16, 44, -48, 6, -48, 48, -16, 57, -62, 18, -56, 54, -2, 72, -64, 20, -60, 60, -24, 62, -64, 8, -64, 84, -24, 68, -72, 24, -96, 72, -8, 74, -76, 31, -80

Offset: 1

Michael Somos, Jul 30 2006

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).

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. A000122, A000700, A010054, A121373.

eta[q_]:= q^(1/24)*QPochhammer[q];  a[n_]:= SeriesCoefficient[(eta[q] *eta[q^3]*eta[q^4]*eta[q^12])^2/(eta[q^2]*eta[q^6])^2, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Mar 07 2018 *)

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

Expansion of (eta(q)*eta(q^3)*eta(q^4)*eta(q^12))^2/(eta(q^2)*eta(q^6))^2 in powers of q.
Euler transform of period 12 sequence [ -2, 0, -4, -2, -2, 0, -2, -2, -4, 0, -2, -4, ...].
Multiplicative with a(2^e) = -(2^e) if e>0, a(3^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) if p>3.
a(3*n)=a(n), a(4n+2)=-2*a(2*n+1).
a(n) = (-1)^(n+1)*A111932(n).
Dirichlet g.f.: (1 - 5/2^s + 1/2^(2*s-2) ) * (1 - 1/3^(s-1)) * zeta(s-1) * zeta(s). - Amiram Eldar, Sep 12 2023

A224976 L.g.f.: log( 1 + Sum_{n>=1} x^(n(3n-1)/2) + x^(n(3n+1)/2) ) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, 1, -2, 1, 6, -8, 8, 1, -11, 6, 12, -20, 14, 8, -12, 1, 18, -35, 20, 6, -16, 12, 24, -44, 31, 14, -38, 8, 30, -48, 32, 1, -24, 18, 48, -83, 38, 20, -28, 6, 42, -64, 44, 12, -66, 24, 48, -92, 57, 31, -36, 14, 54, -116, 72, 8, -40, 30, 60, -120, 62, 32, -88, 1, 84, -96, 68, 18, -48, 48, 72

Offset: 1

Paul D. Hanna, Apr 21 2013

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Compare to: -log( 1 + Sum_{n>=1} (-1)^n*(x^(n*(3*n-1)/2) + x^(n*(3*n+1)/2)) ) = Sum_{n>=1} sigma(n)*x^n/n.

			L.g.f.: A(x) = x + x^2/2 - 2*x^3/3 + x^4/4 + 6*x^5/5 - 8*x^6/6 + 8*x^7/7 + x^8/8 - 11*x^9/9 + 6*x^10/10 + 12*x^11/11 - 20*x^12/12 +...
where
exp(A(x)) = 1 + x + x^2 + x^5 + x^7 + x^12 + x^15 + x^22 + x^26 + x^35 + x^40 + x^51 + x^57 + x^70 + x^77 +...+ x^A001318(n) +...

Paul D. Hanna, Table of n, a(n) for n = 1..10000

Cf. A111932, A001318, A000203 (sigma).

{a(n)=n*polcoeff(log(1+sum(k=1,n,x^(k*(3*k-1)/2) + x^(k*(3*k+1)/2))+x*O(x^n)),n)}
for(n=1,80,print1(a(n),", "))

a(n) = 2*A111932(n) - sigma(n), where sigma(n) is the sum of divisors of n.
a(n) = 1 iff n = 2^k for k>=0.
L.g.f.: log(1 + Sum_{n>=1} x^A001318(n)) = Sum_{n>=1} a(n)*x^n/n, where A001318 are the generalized pentagonal numbers.

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A186100 Expansion of 2 * a(q^2)^2 - a(q)^2 in powers of q where a() is a cubic AGM theta function.

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A131946 Expansion of (phi(-q) * phi(-q^3))^2 in powers of q where phi() is a Ramanujan theta function.

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A232356 Expansion of 2/9 * c(q) * c(q^2) - q * (psi(q) * psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function and c(q) is a cubic AGM theta function.

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A121456 Expansion of q*(psi(-q)*psi(-q^3))^2 in powers of q where psi() is a Ramanujan theta function.

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A224976 L.g.f.: log( 1 + Sum_{n>=1} x^(n*(3*n-1)/2) + x^(n*(3*n+1)/2) ) = Sum_{n>=1} a(n)*x^n/n.

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A121456 Expansion of q(psi(-q)psi(-q^3))^2 in powers of q where psi() is a Ramanujan theta function.

A224976 L.g.f.: log( 1 + Sum_{n>=1} x^(n(3n-1)/2) + x^(n(3n+1)/2) ) = Sum_{n>=1} a(n)*x^n/n.