A030188 -id:A030188 - cp's OEIS frontend

A030209 Expansion of (eta(q) * eta(q^2) * eta(q^3) * eta(q^6))^2 in powers of q.

1, -2, -3, 4, 6, 6, -16, -8, 9, -12, 12, -12, 38, 32, -18, 16, -126, -18, 20, 24, 48, -24, 168, 24, -89, -76, -27, -64, 30, 36, -88, -32, -36, 252, -96, 36, 254, -40, -114, -48, 42, -96, -52, 48, 54, -336, -96, -48, -87, 178, 378, 152, 198, 54, 72, 128

Offset: 1

N. J. A. Sloane

Identical to table 1, p. 493, of Alaca citation. - Jonathan Vos Post, May 24 2007
Unique cusp form of weight 4 for congruence group Gamma_1(6). - Michael Somos, Aug 11 2011
Number 14 of the 74 eta-quotients listed in Table I of Martin (1996).
The table 1, p. 493 of Alaca reference is the first 50 values of c_6(n). - Michael Somos, May 17 2015

			G.f. = q - 2*q^2 - 3*q^3 + 4*q^4 + 6*q^5 + 6*q^6 - 16*q^7 - 8*q^8 + 9*q^9 - 12*q^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Saban Alaca and Kenneth Williams, Evaluation of the convolution sums sum_{l+6m=n} sigma(l)*sigma(m) and sum_{2l+3m=n} sigma(l)*sigma(m), J. Number Theory 124 (2007), no. 2, 491-510. MR2321376 (2008a:11005)
K. Bringmann and K. Ono, Lifting cusp forms to Maass forms with an application to partitions, Proc. Natl. Acad. Sci. USA, 104 (2010), 3725-3731.
Brian Conrey and David Farmer (?), Eta Products and Quotients which are Newforms.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
LMFDB, Space of modular forms of level 6 and weight 4.
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. A030188, A181101, A186100.

Basis( CuspForms( Gamma1(6), 4), 57) [1]; /* Michael Somos, May 17 2015 */

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^3] QPochhammer[ q^6])^2, {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A))^2, n))}; /* Michael Somos, Feb 14 2006 */

CuspForms( Gamma1(6), 4, prec = 57).0; # Michael Somos, Aug 11 2011

Euler transform of period 6 sequence [ -2, -4, -4, -4, -2, -8, ...]. - Michael Somos, Feb 13 2006
a(n) is multiplicative with a(p^e) = (-p)^e if p<5, a(p^e) = a(p) * a(p^(e-1)) - p^3 * a(p^(e-2)) otherwise. - Michael Somos, Feb 13 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 36 (t/i)^4 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 11 2011
G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(6*k)))^2.
a(2*n) = -2 * a(n). Convolution square of A030188. - Michael Somos, May 27 2012
Convolution with A181102 is A186100. - Michael Somos, Jul 07 2015

A159819 Coefficients of L-series for elliptic curve "48a4": y^2 = x^3 + x^2 + x.

Original entry on oeis.org

1, 1, -2, 0, 1, -4, -2, -2, 2, 4, 0, 8, -1, 1, 6, -8, -4, 0, 6, -2, -6, -4, -2, 0, -7, 2, -2, 8, 4, -4, -2, 0, 4, 4, 8, -8, 10, -1, 0, 8, 1, 4, -4, 6, -6, 0, -8, -8, 2, -4, -18, -16, 0, 12, -2, 6, 18, -16, -2, 0, 5, -6, 12, 8, -4, 4, 0, -2, -6, 12, 0, 8, -12

Offset: 0

Michael Somos, Apr 22 2009

sign

Number 54 of the 74 eta-quotients listed in Table I of Martin (1996).
Table I of Martin (1996) for this q-series has exponent of 24 wrong. Number 54 should read 2^(-1)*4^4*6^(-1)*8^(-1)*12^4*24^(-1) (in column g).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
The present expansion corresponds in Martin's notation to
  1^(-1)*2^4*3^(-1)*4^(-1)*6^4*12^(-1). For the expansion of the (corrected) Nr. 54 of Martin's reference see A271231. One finds for the p-defects prime(m) - N(prime(m)) = A271230(m) of the elliptic curve y^2 = x^3 + x^2 + x (mod prime(m)), where N(prime(n)) = A271229(n) is the number of solutions, the modularity pattern A271231(prime(m)) = A271230(m), m >= 1. - Wolfdieter Lang, Apr 18 2016

			G.f. = 1 + x - 2*x^2 + x^4 - 4*x^5 - 2*x^6 - 2*x^7 + 2*x^8 + 4*x^9 + 8*x^11 - ...
G.f. = q + q^3 - 2*q^5 + q^9 - 4*q^11 - 2*q^13 - 2*q^15 + 2*q^17 + 4*q^19 + ...

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. A030188, A271229, A271230, A271231.

A := Basis( CuspForms( Gamma0(48), 2), 147); A[1] + A[3]; /* Michael Somos, Mar 31 2015 */

a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ x^2] QPochhammer[ -x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Mar 31 2015 *)

{a(n) = if(n<0, 0, ellak( ellinit([0, 1, 0, 1, 0], 1), 2*n + 1))};

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

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 2*n+1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p==3, 1, a0=1; a1 = y = -sum(x=0, p-1, kronecker(x^3 + x^2 + x, p)); for(i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1)))};

q='q+O('q^220); Vec( (eta(q^2)*eta(q^6))^4 / (eta(q^1)*eta(q^3)*eta(q^4)*eta(q^12) ) ) \\ Joerg Arndt, Sep 12 2016

Expansion of q^(-1/2) * eta(q^2)^4 * eta(q^6)^4 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)) in powers of q.
Expansion of f(x) * f(-x^2) * f(x^3) * f(-x^6) in powers of x where f() is a Ramanujan theta function.
Euler transform of period 12 sequence [ 1, -3, 2, -2, 1, -6, 1, -2, 2, -3, 1, -4, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 48 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 - (-x)^k) * (1 - x^(2*k)) * (1 - (-x)^(3*k)) * (1 - x^(6*k)).
a(n) = (-1)^n * A030188(n).

A125514 Theta series of 4-dimensional lattice QQF.4.i.

Original entry on oeis.org

1, 4, 20, 4, 52, 24, 20, 32, 116, 4, 120, 48, 52, 56, 160, 24, 244, 72, 20, 80, 312, 32, 240, 96, 116, 124, 280, 4, 416, 120, 120, 128, 500, 48, 360, 192, 52, 152, 400, 56, 696, 168, 160, 176, 624, 24, 480, 192, 244, 228, 620, 72, 728, 216, 20, 288, 928, 80, 600, 240, 312

Offset: 0

N. J. A. Sloane, Jan 31 2007

nonn

			G.f. = 1 + 4*x + 20*x^2 + 4*x^3 + 52*x^4 + 24*x^5 + 20*x^6 + 32*x^7 + 116*x^8 + ...
G.f. = 1 + 4*q^2 + 20*q^4 + 4*q^6 + 52*q^8 + 24*q^10 + 20*q^12 + 32*q^14 + 116*q^16 + ...

John Cannon, Table of n, a(n) for n = 0..5000
G. Nebe and N. J. A. Sloane, Home page for this lattice

Cf. A006353, A030188, A058490, A098098, A123532.

A := Basis(ModularForms( Gamma0(6), 2)); PowerSeries( A[1] + 4*A[2] + 20*A[3], 56); /* Michael Somos, Nov 19 2013 */

a[ n_] := With[{A = QPochhammer[ q] QPochhammer[ q^6], B = QPochhammer[ q^2] QPochhammer[ q^3]}, SeriesCoefficient[ B^7 / A^5 - q A^7 / B^5, {q, 0, n}]] (* Michael Somos, Nov 19 2013 *)

{a(n) = if( n<1, n==0, 2 * qfrep( [ 2, 0, 1, 1; 0, 2, 1, 1; 1, 1, 4, 1; 1, 1, 1, 4 ], n, 1)[n])} /* Michael Somos, May 27 2012 */

{a(n) = local(A, B); if( n<0, 0, A = x * O(x^n); B = eta(x^2 + A) * eta(x^3 + A); A = eta(x + A) * eta(x^6 + A); polcoeff( B^7 / A^5 - x * A^7 / B^5, n))} /* Michael Somos, May 27 2012 */

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 4 * prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 2^(e+2) - 3, if( p==3, 1, (p^(e+1) - 1)/(p - 1))))))} /* Michael Somos, Nov 19 2013 */

A = ModularForms( Gamma0(6), 2, prec=56) . basis(); A[0] + 4*A[1] + 20*A[2]; # Michael Somos, Nov 19 2013

Contribution from Michael Somos, May 27 2012: (Start)
Expansion of (eta(q^2) * eta(q^3))^7 / (eta(q) * eta(q^6))^5 - (eta(q) * eta(q^6))^7 / (eta(q^2) * eta(q^3))^5 in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = + 5*u^4 + 637*v^4 + 1280*w^4 + 352*u^2*w^2 + 342*u^2*v^2 + 5472*v^2*w^2 + 64*u^3*w + 1024*u*w^3 - 68*u^3*v - 756*u*v^3 - 4352*v*w^3 - 3024*v^3*w - 688*u^2*v*w + 2464*u*v^2*w - 2752*u*v*w^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 24 (t/i)^2 f(t) where q = exp(2 Pi i t).
Convolution of A030188 and A058490. a(3*n) = a(n). (End)
a(n) = 4*b(n) where b(n) is multiplicative and b(2^e) = 2^(e+2) - 3, b(3^e) = 1, b(p^e) = (p^(e+1) - 1)/(p - 1) otherwise. - Michael Somos, Nov 19 2013
a(n) = A006353(n) - A123532(n). a(6*n + 5) = 24 * A098098(n). - Michael Somos, Nov 19 2013

A246707 Expansion of phi(-q) * phi(-q^2) * phi(-q^3) * phi(-q^6) in powers of q.

Original entry on oeis.org

1, -2, -2, 2, 6, 4, -14, 0, 6, -2, -12, -8, 42, 4, -16, -4, 6, -4, -50, 8, 36, 0, -24, 16, 42, 2, -28, 2, 48, -12, -84, -16, 6, 8, -36, 0, 150, -12, -40, -4, 36, 12, -112, -8, 72, 4, -48, 0, 42, 14, -62, 4, 84, 4, -158, 16, 48, -8, -60, -8, 252, 4, -64, 0, 6

Offset: 0

Michael Somos, Sep 01 2014

sign

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

			G.f. = 1 - 2*q - 2*q^2 + 2*q^3 + 6*q^4 + 4*q^5 - 14*q^6 + 6*q^8 - 2*q^9 + ...

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. A030188, A033765.

A := Basis( ModularForms( Gamma0(24), 2), 26); A[1] - 2*A[2] - 2*A[3] + 2*A[4] + 6*A[5] + 4*A[6] - 14*A[7] + 6*A[8];

eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q]^2* eta[q^2]*eta[q^3]^2*eta[q^6]/(eta[q^4]*eta[q^12]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 18 2018 *)

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

q='q+O('q^99); Vec(eta(q)^2*eta(q^2)*eta(q^3)^2*eta(q^6)/(eta(q^4)*eta(q^12))) \\ Altug Alkan, Apr 18 2018

Expansion of eta(q)^2 * eta(q^2) * eta(q^3)^2 * eta(q^6) / (eta(q^4) * eta(q^12)) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 384 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033765.
a(2*n + 1) = -2 * A030188(n).

A258090 Expansion of q^(-5/6) * (eta(q) * eta(q^6)^2 / eta(q^3))^2 in powers of q.

Original entry on oeis.org

1, -2, -1, 4, -3, 0, 3, 0, 1, -2, -2, -4, 0, 2, 3, -4, 9, 6, -9, 0, -6, 2, 3, 4, -7, 8, 0, -12, -3, -6, 6, 0, 9, 0, 8, 4, 2, -6, -5, 8, -7, -10, -1, 4, 5, 2, -13, 0, 9, -8, -2, 12, -3, 4, 0, -4, -16, 6, -1, 12, 10, 0, 6, 0, 1, -8, 15, -12, 0, -6, 1, -16, -16

Offset: 0

Michael Somos, May 19 2015

sign

			G.f. = 1 - 2*x - x^2 + 4*x^3 - 3*x^4 + 3*x^6 + x^8 - 2*x^9 - 2*x^10 + ...
G.f. = q^5 - 2*q^11 - q^17 + 4*q^23 - 3*q^29 + 3*q^41 + q^53 - 2*q^59 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A030188.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^6]^2 / QPochhammer[ x^3])^2, {x, 0, n}];

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

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 6*n + 5; A = factor(n); -1/2 * prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p==3, (-1)^e, a0 = 1; a1 = y = -sum( x=0, p-1, kronecker( x^3 - x^2 - 4*x + 4, p)); for( i=2, e, x = y*a1 - p*a0; a0 = a1; a1 = x); a1)))};

q='q+O('q^99); Vec((eta(q)*eta(q^6)^2/eta(q^3))^2) \\ Altug Alkan, Aug 02 2018

Euler transform of period 6 sequence [ -2, -2, 0, -2, -2, -4, ...].
G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(3*k))^2 * (1 + x^(3*k))^4.
-2 * a(n) = A030188(3*n + 2).

A258065 Expansion of (phi(-x^3) * f(-x^2))^2 in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 0, -2, -4, -1, 8, 6, 4, -7, -8, -2, -4, 10, -8, -4, 0, 2, 16, -2, 16, 5, -8, 0, -12, -12, -16, -2, 12, -9, 0, 6, 8, 2, 16, 12, -20, 0, -8, 22, 0, 18, 8, -32, 0, 4, 8, -26, -28, -13, -8, 0, 12, -6, 24, 2, 20, 18, 0, 30, -16, -3, -8, -10, 20, 0, -16, 14, -16

Offset: 0

Michael Somos, May 18 2015

sign

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

			G.f. = 1 - 2*x^2 - 4*x^3 - x^4 + 8*x^5 + 6*x^6 + 4*x^7 - 7*x^8 - 8*x^9 + ...
G.f. = q - 2*q^13 - 4*q^19 - q^25 + 8*q^31 + 6*q^37 + 4*q^43 - 7*q^49 + ...

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

A := Basis( ModularForms( Gamma0(72), 2), 409); A[2] - 2*A[14];

A := Basis( CuspForms( Gamma0(72), 2), 409); A[1];

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] QPochhammer[ x^2])^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^3]^2 / QPochhammer[ x^6])^2, {x, 0, n}];

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

Expansion of (f(x, x^2) * f(-x))^2 in powers of x where f() is the Ramanujan general theta function.
Expansion of q^(-1/6) * (eta(q^2) * eta(q^3)^2 / eta(q^6))^2 in powers of q.
Euler transform of period 6 sequence [ 0, -2, -4, -2, 0, -4, ...].
a(n) = A030188(3*n).

A276847 Expansion of eta(q^2) * eta(q^4) * eta(q^6) * eta(q^12) in powers of q.

Original entry on oeis.org

1, 0, -1, 0, -2, 0, 0, 0, 1, 0, 4, 0, -2, 0, 2, 0, 2, 0, -4, 0, 0, 0, -8, 0, -1, 0, -1, 0, 6, 0, 8, 0, -4, 0, 0, 0, 6, 0, 2, 0, -6, 0, 4, 0, -2, 0, 0, 0, -7, 0, -2, 0, -2, 0, -8, 0, 4, 0, 4, 0, -2, 0, 0, 0, 4, 0, -4, 0, 8, 0, 8, 0, 10, 0, 1, 0, 0, 0, -8, 0, 1, 0

Offset: 1

Seiichi Manyama, Sep 22 2016

The bisection of this sequence containing all nonzero terms is A030188.

Multiplicative. See A030188 for formula. - Andrew Howroyd, Jul 31 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

Cf. A030188, A271231, A276807, A276649.

CoefficientList[Series[QPochhammer[x^2] QPochhammer[x^4] QPochhammer[x^6] QPochhammer[x^12], {x, 0, 100}], x] (* Jan Mangaldan, Jan 04 2017 *)

a(4*n-3) = A271231(4*n-3), a(4*n-2) = 0, a(4*n-1) = -A271231(4*n-1), a(4*n) = 0.
G.f.: x * Product_{k>0} (1 - x^(2*k)) * (1 - x^(4*k)) * (1 - x^(6*k)) * (1 - x^(12*k)).
a(2*n+1) = A030188(n). - Michel Marcus, Sep 25 2016
Euler transform of period 12 sequence [0, -1, 0, -2, 0, -2, 0, -2, 0, -1, 0, -4, ...]. - Georg Fischer, Nov 17 2022

cp's OEIS Frontend

A030209 Expansion of (eta(q) * eta(q^2) * eta(q^3) * eta(q^6))^2 in powers of q.

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A159819 Coefficients of L-series for elliptic curve "48a4": y^2 = x^3 + x^2 + x.

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A125514 Theta series of 4-dimensional lattice QQF.4.i.

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A246707 Expansion of phi(-q) * phi(-q^2) * phi(-q^3) * phi(-q^6) in powers of q.

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A258090 Expansion of q^(-5/6) * (eta(q) * eta(q^6)^2 / eta(q^3))^2 in powers of q.

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A258065 Expansion of (phi(-x^3) * f(-x^2))^2 in powers of x where phi(), f() are Ramanujan theta functions.

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