A152243 -id:A152243 - cp's OEIS frontend

A030208 Expansion of q^(-1/2) * (eta(q) * eta(q^3))^3 in powers of q.

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

N. J. A. Sloane

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Number 28 of the 74 eta-quotients listed in Table I of Martin (1996).

Expansion of a newform level 12 weight 3 and character [0, 1].

			G.f. = 1 - 3*x + 2*x^3 + 9*x^4 - 22*x^6 + 26*x^9 - 6*x^10 + 25*x^12 - 27*x^13 + ...
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
Author?, Eta Products and Quotients which are Newforms. [Broken link?]
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. A007332, A152243, A209939.

A := Basis( CuspForms( Gamma1(12), 3), 140); A[1] - 3*A[3]; /* Michael Somos, May 17 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^3])^3, {x, 0, n}]; (* Michael Somos, May 17 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^3, n))}; /* Michael Somos, Jun 14 2007 */

Euler transform of period 3 sequence [ -3, -3, -6, ...]. - Michael Somos, Feb 13 2006
a(n) = b(2*n + 1) where b() is multiplicative with 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. - Michael Somos, Feb 13 2006
G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(3*k)))^3.
G.f.: Sum_{k>=0} a(k) * q^(2*k + 1) = (1/2) * Sum_{u, v in Z} (u*u - 3*v*v) * q^(u*u + 3*v*v). - Michael Somos, Jun 14 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Nov 16 2008
a(3*n + 1) = -3 * a(n). a(3*n + 2) = 0. a(3*n) = A152243(n). - Michael Somos, Mar 09 2012
a(n) = (-1)^n * A209939(n). - Michael Somos, Mar 16 2012
Convolution square is A007332. - Michael Somos, Nov 16 2008

A153728 Expansion of q^(-1/3) * (eta(q)^8 + 8 * eta(q^4)^8) in powers of q^2.

Original entry on oeis.org

1, 20, -70, 56, -125, 308, 110, -520, 57, 0, 182, -880, 1190, 884, 0, -1400, -1330, 1820, -646, 0, -1331, 380, 1120, 2576, 0, 1748, -3850, -3400, 2703, -2500, 3458, 0, -1150, -5236, 0, 6032, 6160, -3220, 4466, 0, -7378, -3920, 0, 2200, 0, 812, -4030, 5600, -4913

Offset: 0

Michael Somos, Dec 31 2008

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This is a member of an infinite family of integer weight modular forms. g_1 = A097195, g_2 = A000727, g_3 = A152243, g_4 = A153728. - Michael Somos, Jun 10 2015

			G.f. = 1 + 20*x - 70*x^2 + 56*x^3 - 125*x^4 + 308*x^5 + 110*x^6 - 520*x^7 + ...
G.f. = q + 20*q^7 - 70*q^13 + 56*q^19 - 125*q^25 + 308*q^31 + 110*q^37 + ...

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

Cf. A000727, A000731, A097195, A152243, A153729, A161969.

A := Basis( CuspForms( Gamma0(36), 4), 289); A[1] + 20*A[7] - 70*A[12]; /* Michael Somos, Jun 10 2015 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^8 + 8 x QPochhammer[ x^4]^8, {x, 0, 2 n}]; (* Michael Somos, Jun 10 2015 *)

{a(n) = my(A); if( n<0, 0, n *= 2; A = x * O(x^n); polcoeff( eta(x + A)^8 + 8 * x * eta(x^4 + A)^8, n))};

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 6*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%6==5, if( e%2, 0, (-p)^(3*e/2)), for(x=1, sqrtint(p\3), if( issquare(p-3*x^2, &y), break)); if( y%3!=1, y=-y); y*=2; y = y^3 - 3*p*y; a0=1; a1=y; for(i=2, e, x = y*a1 - p^3*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Jun 10 2015 */

a(n) = b(6*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * (-1)^(e/2) * p^(3*e/2) if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - b(p^(e-2)) * p^3 if p == 1 (mod 6) where b(p) = (x^2 - 3*p)*x, 4*p = x^2 + 3*y^2, |x| < |y| and x == 2 (mod 3).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 648 (t/i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A153729.
a(n) = A000731(2*n) = A153729(2*n) = A161969(2*n). - Michael Somos, Jun 10 2015

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

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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.

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A030208 Expansion of q^(-1/2) * (eta(q) * eta(q^3))^3 in powers of q.

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A153728 Expansion of q^(-1/3) * (eta(q)^8 + 8 * eta(q^4)^8) in powers of q^2.

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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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