A133089 -id:A133089 - cp's OEIS frontend

A133079 Expansion of f(x)^3 - 3 * x * f(x^9)^3 in powers of x^3 where f() is a Ramanujan theta function.

1, -5, -7, 0, 0, 11, 0, -13, 0, 0, 0, 0, 17, 0, 0, 19, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, -29, 0, 0, 0, 0, -31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, -37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, 43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -47, 0, 0, 0

Offset: 0

Michael Somos, Sep 08 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).
There is a plus sign on the left side and the first and third plus signs on the right side which should be minuses in Ramanujan's equation.

			G.f. = 1 - 5*x - 7*x^2 + 11*x^5 - 13*x^7 + 17*x^12 + 19*x^15 - 23*x^22 + ...
G.f. = q - 5*q^25 - 7*q^49 + 11*q^121 - 13*q^169 + 17*q^289 + 19*q^361 - ...

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 357, Entry 5, Eq. (5.1)
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 266.

Seiichi Manyama, 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. A116916, A133089, A204850.

a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ@m, m (-1)^Boole[Mod[m, 8] > 4], 0]]; (* Michael Somos, Jun 19 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^3 - 3 x QPochhammer[ -x^9]^3, {x, 0, 3 n}]; (* Michael Somos, Jun 19 2015 *)

{a(n) = if( issquare( 24*n + 1, &n), n * (-1) ^ (n%8 > 4), 0)};

{a(n) = my(A, p, e); if( n<0, 0, n = 24*n + 1; A = factor(n); prod(k = 1, matsize(A) [1], [p, e] = A[k,]; if( p < 5, 0, p *= kronecker( -2, p); if( e%2, 0, p^(e/2) ))))};

{a(n) = my(A); if( n<0, 0, n *= 3; A = x * O(x^n); polcoeff( eta(-x + A)^3 - 3 * x * eta(-x^9 + A)^3, n))};

Expansion of f(x) * a(-x) in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = -192 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A204850.
a(n) = b(24*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(e/2) if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 5, 7 (mod 8).
G.f.: Sum_{k in Z} Kronecker( 2, 2*k + 1) * (6*k + 1) * x^(k * (3*k + 1)/2).
a(5*n + 3) = a(5*n + 4) = 0. a(25*n + 1) = -5 * a(n). a(n) = (-1)^n * A116916(n).
a(n) = A133089(3*n) = A204850(3*n). - Michael Somos, Jun 19 2015

A209676 Expansion of f(x)^12 in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 12, 54, 88, -99, -540, -418, 648, 594, -836, 1056, 4104, -209, -4104, -594, -4256, -6480, 4752, -298, -5016, 17226, 12100, -5346, 1296, -9063, 7128, 19494, -29160, -10032, 7668, -34738, -8712, -22572, -21812, 49248, 46872, 67562, -2508, -47520, 76912

Offset: 0

Michael Somos, Mar 11 2012

sign

Number 35 of the 74 eta-quotients listed in Table I of Martin (1996). See g.f. B(q) below: cusp form weight 6 level 16.

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

			G.f. = 1 + 12*x + 54*x^2 + 88*x^3 - 99*x^4 - 540*x^5 - 418*x^6 + 648*x^7 + ...
G.f. B(q) of {b(n)}: q + 12*q^3 + 54*q^5 + 88*q^7 - 99*q^9 - 540*q^11 - 418*q^13 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A121373, A133089, A187076.

A000735 is the same except for signs.

A := Basis( CuspForms( Gamma0(16), 6), 81); A[1] + 12*A[3] + 54*A[5] + 88*A[7]; /* Michael Somos, Jun 09 2015 */

a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^12, {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)

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

Expansion of q^(-1/2) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^12 in powers of q.
Euler transform of period 4 sequence [ 12, -24, 12, -12, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^5 * b(p^(e-2)) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 4096 (t/i)^6 f(t) where q = exp(2 Pi i t).
G.f.: (Product_{k>0} (1 - (-x)^k))^12.
a(n) = (-1)^n * A000735(n).
Convolution cube of A187076. Convolution fourth power of A133089. Convolution twelfth power of A121373.

A204850 Expansion of f(x)^3 - 9 * x * f(x^9)^3 in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, -6, 0, -5, 0, 0, -7, 0, 0, 0, -18, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, -13, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0

Offset: 0

Michael Somos, Jan 19 2012

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 - 6*x - 5*x^3 - 7*x^6 - 18*x^10 + 11*x^15 - 13*x^21 + 30*x^28 + ...
G.f. = q - 6*q^9 - 5*q^25 - 7*q^49 - 18*q^81 + 11*q^121 - 13*q^169 + ...

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. A133079, A133089, A202394.

a[ n_] := With[ {m = Sqrt[8 n + 1]}, If[ IntegerQ@m, m (-1)^(n + Quotient[m, 6]), 0] If[ Divisible[ m, 3], 2, 1]]; (* Michael Somos, Jun 19 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^3 - 9 x QPochhammer[ -x^9]^3, {x, 0, n}]; (* Michael Somos, Jun 19 2015 *)

{a(n) = my(m); if( issquare(8*n + 1, &m), (-1)^(m \ 6 + n) * m * ((m%3 == 0) + 1), 0)};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(-x + A)^3 - 9 * x * eta(-x^9 + A)^3, n))};

Expansion of f(x^3) * a(-x) in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM theta function.
a(n) = b(8*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = - (1 + (-1)^e) * 3^(e/2), b(p^e) = (1 + (-1)^e)/2 * p^(e/2) if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 5, 7 (mod 8). - Michael Somos, Jun 19 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = -576 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133079.
a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = a(9*n + 4) = a(9*n + 7) = 0. a(3*n) = A133079(n). a(9*n + 1) = -6 * A133089(n). a(25*n + 3) = -5 * a(n). a(n) = (-1)^n * A202394(n).

A209941 Expansion of f(x)^6 in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 6, 9, -10, -30, 0, 11, -42, 0, 70, 18, 54, 49, -90, 0, 22, -60, 0, -110, 0, 81, -180, -78, 0, 130, 198, 0, 182, -30, -90, 121, -84, 0, 0, 210, 0, -252, 102, -270, -170, 0, 0, -69, -330, 0, 38, 420, 0, -190, 390, 0, 108, 0, 0, 0, 300, 99, -442, 210, 0, 418

Offset: 0

Michael Somos, Mar 16 2012

sign

Number 59 of the 74 eta-quotients listed in Table I of Martin (1996).

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

			G.f. = 1 + 6*x + 9*x^2 - 10*x^3 - 30*x^4 + 11*x^6 - 42*x^7 + 70*x^9 + ...
G.f. = q + 6*q^5 + 9*q^9 - 10*q^13 - 30*q^17 + 11*q^25 - 42*q^29 + 70*q^37 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000729, A133089, A208845.

seq(coeff(series(mul((1-(-x)^k)^6,k=1..n), x,n+1),x,n),n=0..70); # Muniru A Asiru, Aug 12 2018

a[ n_] := SeriesCoefficient[QPochhammer[ x^2]^18 / (QPochhammer[ x] QPochhammer[ x^4])^6, {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)
CoefficientList[Series[(QPochhammer[x^2]^3/(QPochhammer[x] QPochhammer[x^4]))^6, {x, 0, 50}], x] (* G. C. Greubel, Aug 11 2018 *)

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

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, A = factor(4*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p%4==3, if( e%2, 0, p^e), for( i=1, sqrtint(p\2), if( issquare(p - i^2, &y), x=i; break)); a0=1; a1 = y = 2 * (x^2 - y^2) * (-1)^((p%8==5) + y); for( i=2, e, x = y*a1 - p^2*a0; a0=a1; a1=x); a1)))};

Expansion of q^(-1/4) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^6 in powers of q.
Euler transform of period 4 sequence [6, -12, 6, -6, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 512 (t/i)^3 f(t) where q = exp(2 Pi i t).
a(n) = b(4*n + 1) where b(n) is multiplicative b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^e if p == 3 (mod 4), else b(p^e) = b(p) * b(p^(e-1)) - p^2 * b^(p^(e-2)) if p == 1 (mod 4).
G.f.: Product_{k>0} (1 - (-x)^k)^6.
a(n) = (-1)^n * A000729(n). a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = 9 * a(n).
Convolution square of A133089. Convolution cube of A208845. - Michael Somos, Jun 09 2015

cp's OEIS Frontend

A133079 Expansion of f(x)^3 - 3 * x * f(x^9)^3 in powers of x^3 where f() is a Ramanujan theta function.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A209676 Expansion of f(x)^12 in powers of x where f() is a Ramanujan theta function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A204850 Expansion of f(x)^3 - 9 * x * f(x^9)^3 in powers of x where f() is a Ramanujan theta function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A209941 Expansion of f(x)^6 in powers of x where f() is a Ramanujan theta function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula