A030208 -id:A030208 - cp's OEIS frontend

A007332 Expansion of 6-dimensional cusp form (eta(q) * eta(q^3))^6 in powers of q.

0, 1, -6, 9, 4, 6, -54, -40, 168, 81, -36, -564, 36, 638, 240, 54, -1136, 882, -486, -556, 24, -360, 3384, -840, 1512, -3089, -3828, 729, -160, 4638, -324, 4400, 1440, -5076, -5292, -240, 324, -2410, 3336, 5742, 1008, -6870, 2160, 9644, -2256, 486, 5040

Offset: 0

N. J. A. Sloane, Mira Bernstein

Number 5 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = q - 6*q^2 + 9*q^3 + 4*q^4 + 6*q^5 - 54*q^6 - 40*q^7 + 168*q^8 + 81*q^9 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 204.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 145, problem 13.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
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

Cf. A030208.

Basis( CuspForms( Gamma0(3), 6), 47) [1]; /* Michael Somos, Dec 10 2013 */

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^3] )^6, {q, 0, n}]; (* Michael Somos, May 28 2013 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^6, n))}; /* Michael Somos, Jul 16 2004 */

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ( prod( k=1, n, (1 - (k%3==0) * x^k) * (1 - x^k), 1 + A) )^6, n))}; /* Michael Somos, Jul 16 2004 */

CuspForms( Gamma0(3), 6, prec=47).0; # Michael Somos, May 28 2013

G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(3*k)))^6.
Expansion of (eta(q) * eta(q^3))^6 in powers of q. - Michael Somos, Jul 16 2004
Euler transform of period 3 sequence [ -6, -6, -12, ...]. - Michael Somos, Jul 16 2004
Expansion of a newform of level 3, weight 6 and trivial character. - Michael Somos, Nov 16 2008
a(n) is multiplicative with a(3^e) = 9^e, a(p^e) = a(p) * a(p^(e-1)) - p^5 * a(p^(e-2)). - Michael Somos, Mar 08 2006
Given A = A0 + A1 + A2 is the 3-section, then 0 = A2^2 - 4 * A1*A0. - Michael Somos, Mar 08 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u * w * (u + 12 * v + 64 * w) - v^3. - Michael Somos, May 02 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 3^3 (t/i)^6 f(t) where q = exp(2 Pi i t). - Michael Somos, Nov 16 2008
a(3*n) = 9 * a(n). - Michael Somos, Nov 16 2008
Convolution square of A030208.

A152243 Expansion of a(q) * f(-q)^4 where f() is a Ramanujan theta function and a() is a cubic AGM function.

Original entry on oeis.org

1, 2, -22, 26, 25, -46, 26, -22, -45, 0, 74, 122, -46, -142, 0, -44, 2, 194, -214, 0, 121, 146, 52, -22, 0, -286, -118, -262, 315, 50, 314, 0, -382, 386, 0, -166, -92, 338, 26, 0, -286, -572, 0, 52, 0, 242, 122, 458, 289, 0, -44, -358, -142, 0, -550, 362, 482, -188, -502, 0, 315, -718, 698, -694

Offset: 0

Michael Somos, Nov 30 2008

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 + 2*x - 22*x^2 + 26*x^3 + 25*x^4 - 46*x^5 + 26*x^6 - 22*x^7 + ...
G.f. = q + 2*q^7 - 22*q^13 + 26*q^19 + 25*q^25 - 46*q^31 + 26*q^37 - 22*q^43 + ...

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. A030208, A152244.

eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-1/6)*((eta[q^(1/3)]*eta[q^1])^3 + 3*(eta[q^1]*eta[q^3])^3), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 10 2018 *)

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

Expansion of q^(-1/2) * ( (eta(q) * eta(q^3))^3 + 3 * (eta(q^3) * eta(q^9))^3 ) in powers of q^3.
a(n) = b(6*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = p^e * (1 + (-1)^e) / 2 if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - p^2 * b(p^(e-2)) if p == 1 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 15552^(1/2) (t / i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A152244.
a(n) = A030208(3*n) = A152244(3*n).

A152244 Expansion of a(x) * f(-x^3)^4 in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM function.

Original entry on oeis.org

1, 6, 0, 2, -18, 0, -22, 0, 0, 26, 12, 0, 25, 54, 0, -46, 0, 0, 26, -132, 0, -22, 0, 0, -45, 0, 0, 0, 156, 0, 74, -36, 0, 122, 0, 0, -46, 150, 0, -142, -162, 0, 0, 0, 0, -44, -276, 0, 2, 0, 0, 194, 0, 0, -214, 156, 0, 0, 396, 0, 121, 0, 0, 146, -132, 0, 52, 0, 0, -22, 0, 0, 0, -270, 0, -286, 0, 0, -118

Offset: 0

Michael Somos, Nov 30 2008

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 + 2*x^3 - 18*x^4 - 22*x^6 + 26*x^9 + 12*x^10 + 25*x^11 + ...
G.f. = q + 6*q^3 + 2*q^7 - 18*q^9 - 22*q^13 + 26*q^19 + 12*q^21 + 25*q^25 + ...

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. A030208, A152243.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^3])^3 + 9 x (QPochhammer[ x^3] QPochhammer[ x^9])^3 , {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)

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

Expansion of q^(-1/2) *( (eta(q)*eta(q^3))^3 + 9*(eta(q^3)*eta(q^9))^3 ) in powers of q.
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = -2 * (-3)^e if e>0, b(p^e) = p^e * (1 + (-1)^e) / 2 if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - p^2 * b(p^(e-2)) if p == 1 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 139968^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A152243.
a(3*n) = A152243(n). a(3*n + 1) = 6 * A030208(n). a(3*n + 2) = 0.

A209939 Expansion of (f(x) * f(x^3))^3 in powers of q where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 3, 0, -2, 9, 0, -22, 0, 0, -26, -6, 0, 25, 27, 0, 46, 0, 0, 26, -66, 0, 22, 0, 0, -45, 0, 0, 0, -78, 0, 74, -18, 0, -122, 0, 0, -46, 75, 0, 142, 81, 0, 0, 0, 0, 44, 138, 0, 2, 0, 0, -194, 0, 0, -214, 78, 0, 0, -198, 0, 121, 0, 0, -146, 66, 0, 52, 0, 0, 22

Offset: 0

Michael Somos, Mar 15 2012

sign

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

Number 55 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 + 3*x - 2*x^3 + 9*x^4 - 22*x^6 - 26*x^9 - 6*x^10 + 25*x^12 + ...
G.f. = q + 3*q^3 - 2*q^7 + 9*q^9 - 22*q^13 - 26*q^19 - 6*q^21 + 25*q^25 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
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. A030208, A208978.

a[ n_] := SeriesCoefficient[ (QPochhammer[ -x] QPochhammer[ -x^3])^3, {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) * eta(x^6 + A))^3 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)))^3, n))};

Expansion of q^(-1/2) * ((eta(q^2) * eta(q^6))^3 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)))^3 in powers of q.
Euler transform of period 12 sequence [3, -6, 6, -3, 3, -12, 3, -3, 6, -6, 3, -6, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 48^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 3^e, b(p^e) = (1 + (-1)^e) / 2 * p^e if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - p^2 * b(p^(e-2)) otherwise.
G.f.: ( Product_{k>0} (1 - (-x)^k) * (1 - (-x)^(3*k)) )^3.
a(n) = (-1)^n * A030208(n). a(3*n + 2) = 0. a(3*n + 1) = 3 * a(n).
Conovlution cube of A208978. - Michael Somos, Jun 09 2015

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A007332 Expansion of 6-dimensional cusp form (eta(q) * eta(q^3))^6 in powers of q.

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A152243 Expansion of a(q) * f(-q)^4 where f() is a Ramanujan theta function and a() is a cubic AGM function.

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A152244 Expansion of a(x) * f(-x^3)^4 in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM function.

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A209939 Expansion of (f(x) * f(x^3))^3 in powers of q where f() is a Ramanujan theta function.

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