A000731 -id:A000731 - cp's OEIS frontend

A130539 Expansion of q^(-1/3) * a(q) * b(q) * c(q) / 3 in powers of q where a(), b(), c() are cubic AGM theta functions.

1, 4, -13, 0, -1, 16, 11, 0, 25, -52, -46, 0, 47, 0, -22, 0, 120, -4, 0, 0, -121, 64, -109, 0, -97, 44, 131, 0, 0, 0, 13, 0, 167, 100, -37, 0, -214, -208, 0, 0, 121, -184, 146, 0, -143, 0, 251, 0, 0, 188, 59, 0, -118, 0, 299, 0, -168, -88, -325, 0, -313

Offset: 0

Michael Somos, Jun 03 2007

sign

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Denoted by g_3(q) in Cynk and Hulek in Remark 3.4 on page 12 as the unique level 27 form of weight 3.
This is a member of an infinite family of integer weight modular forms. g_1 = A033687, g_2 = A030206, g_3 = A130539, g_4 = A000731.
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).

			G.f. = 1 + 4*x - 13*x^2 - x^4 + 16*x^5 + 11*x^6 + 25*x^8 - 52*x^9 - 46*x^10 + ...
G.f. = q + 4*q^4 - 13*q^7 - q^13 + 16*q^16 + 11*q^19 + 25*q^25 - 52*q^28 - ...

S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds

Cf. A000731, A030206, A033687.

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^3] (QPochhammer[ x]^3 + 9 x QPochhammer[ x^9]^3), {x, 0, n}]; (* Michael Somos, Oct 20 2015 *)

{a(n) = my(A, p, e, x, y, a0, a1); n = 3*n + 1; if( n<1, 0, A=factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, if( p%3==2, if( e%2, 0, p^e), for( x=1, sqrtint(4*p\27), if( issquare(4*p - 27*x^2, &y), break)); y = y^2 - p*2; a0=1; a1=y; for( i=2, e, x=y*a1 - p^2*a0; a0=a1; a1=x); a1))))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^3 + A) * (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3), n))};

Expansion of q^(-1/3) * ( eta(q)^5 * eta(q^3) + 9 * eta(q)^2 * eta(q^3) * eta(q^9)^3 ) in powers of q.
a(n) = b(3*n + 1) where b() is multiplicative and b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^e if p == 2 (mod 3), b(p^e) = b(p) * b(p^(e-1)) - p^2 * b(p^(e-2)) if p == 1 (mod 3) where b(p) = x^2 - 2*p, 4*p = x^2 + 3*y^2, |x|<|y| and x == 2 (mod 3).
G.f.: Sum_{k>=0} a(k) * x^(3*k + 1) = (1/2) * Sum_{u, v in Z} (u*u - 7*v*v) * x^(u*u + u*v + 7*v*v). - Michael Somos, Jun 14 2007
a(4*n + 1) = 4*a(n). a(4*n + 3) = 0. - Michael Somos, Oct 20 2015

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

sign

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

A092342 a(n) = sigma_3(3n+1).

Original entry on oeis.org

1, 73, 344, 1134, 2198, 4681, 6860, 11988, 15751, 25112, 29792, 44226, 50654, 73710, 79508, 109512, 117993, 160454, 167832, 219510, 226982, 299593, 300764, 390096, 389018, 500780, 493040, 620298, 619164, 779220, 756112, 934416, 912674, 1149823, 1092728

Offset: 0

N. J. A. Sloane, Mar 20 2004

			q + 73*q^4 + 344*q^7 + 1134*q^10 + 2198*q^13 + 4681*q^16 + ...

Trisection of A001158.

Cf. A000731, A013662, A033690, A045823, A091986, A092341, A092343.

DivisorSigma[3,3*Range[0,40]+1] (* Harvey P. Dale, Apr 22 2019 *)

{a(n) = if(n<0, 0, sigma(3*n+1, 3))} /* Michael Somos, Aug 22 2007 */

Expansion of q^(-1/3) * c(q) * (c(q)^3 + b(q)^3 / 3) in powers of q where b(), c() are cubic AGM functions. - Michael Somos, Aug 22 2007
If b(3*n) = 0, b(3*n+1) = a(n), b(3*n+2) = A092343(n), then b(n) is multiplicative with b(3^e) = 0^e, b(p^e) = (p^(3*e+3) - 1) / (p^3 - 1) otherwise. - Michael Somos, Aug 22 2007
a(n) = A000731(n) + 81*A033690(n-1). - Michael Somos, Aug 22 2007
Sum_{k=0..n} a(k) ~ (20*zeta(4)/3) * n^4. - Amiram Eldar, Dec 12 2023

A319933 A(n, k) = [x^k] DedekindEta(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, -1, -2, 1, 0, 0, -1, -3, 1, 0, 0, 2, 0, -4, 1, 0, 1, 1, 5, 2, -5, 1, 0, 0, 2, 0, 8, 5, -6, 1, 0, 1, -2, 0, -5, 10, 9, -7, 1, 0, 0, 0, -7, -4, -15, 10, 14, -8, 1, 0, 0, -2, 0, -10, -6, -30, 7, 20, -9, 1, 0, 0, -2, 0, 8, -5, 0, -49, 0, 27, -10, 1

Offset: 0

Peter Luschny, Oct 02 2018

The columns are generated by polynomials whose coefficients constitute the triangle of signed D'Arcais numbers A078521 when multiplied with n!.

			[ 0] 1,   0,   0,    0,     0,    0,     0,     0,     0,     0, ... A000007
[ 1] 1,  -1,  -1,    0,     0,    1,     0,     1,     0,     0, ... A010815
[ 2] 1,  -2,  -1,    2,     1,    2,    -2,     0,    -2,    -2, ... A002107
[ 3] 1,  -3,   0,    5,     0,    0,    -7,     0,     0,     0, ... A010816
[ 4] 1,  -4,   2,    8,    -5,   -4,   -10,     8,     9,     0, ... A000727
[ 5] 1,  -5,   5,   10,   -15,   -6,    -5,    25,    15,   -20, ... A000728
[ 6] 1,  -6,   9,   10,   -30,    0,    11,    42,     0,   -70, ... A000729
[ 7] 1,  -7,  14,    7,   -49,   21,    35,    41,   -49,  -133, ... A000730
[ 8] 1,  -8,  20,    0,   -70,   64,    56,     0,  -125,  -160, ... A000731
[ 9] 1,  -9,  27,  -12,   -90,  135,    54,   -99,  -189,   -85, ... A010817
[10] 1, -10,  35,  -30,  -105,  238,     0,  -260,  -165,   140, ... A010818
    A001489,  v , A167541, v , A319931,  v ,         diagonal: A008705
           A080956       A319930      A319932

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Clarendon Press, Oxford, 2003.

M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Vaclav Kotesovec, The integration of q-series
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
M. Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
Tim Silverman, Counting Cliques in Finite Distant Graphs, arXiv preprint arXiv:1612.08085 [math.CO], 2016.
Michael Somos, Introduction to Ramanujan theta functions
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Transpose of A286354.

Cf. A078521, A319574 (JacobiTheta3).

# DedekindEta is defined in A000594
for n in 0:10
    DedekindEta(10, n) |> println
end

DedekindEta := (x, n) -> mul(1-x^j, j=1..n):
A319933row := proc(n, len) series(DedekindEta(x, len)^n, x, len+1):
seq(coeff(%, x, j), j=0..len-1) end:
seq(print([n], A319933row(n, 10)), n=0..10);

eta[x_, n_] := Product[1 - x^j, {j, 1, n}];
A[n_, k_] := SeriesCoefficient[eta[x, k]^n, {x, 0, k}];
Table[A[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 10 2018 *)

from sage.modular.etaproducts import qexp_eta
def A319933row(n, len):
    return (qexp_eta(ZZ['q'], len+4)^n).list()[:len]
for n in (0..10):
    print(A319933row(n, 10))

A062248 Expansion of a Schwarzian ({f_{27|3}, tau} / (4*Pi)^2) in powers of q^3.

Original entry on oeis.org

1, -48, -216, 1536, -1560, -3024, 13824, -8736, -14040, 41712, -27216, -31968, 112128, -51072, -74304, 193536, -113880, -117936, 375408, -165984, -220752, 528384, -287712, -292032, 898560, -375024, -474768, 1126464, -598848, -585360, 1741824, -722400, -898776

Offset: 0

N. J. A. Sloane, Jul 01 2001

sign

The q-series f_{27|3} is the g.f. for A062246. This is given on page 274 of McKay and Sebbar along with equation (8.2) which gives an expression for the g.f. A(q) of this sequence, but the left side is A(q^3) and the right side is A(q). - Michael Somos, Aug 12 2014
Ramanujan theta function: f(-q) (see A010815). Ramanujan Lambert series: Q(q) = E_4(q) (see A004009).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 48*x - 216*x^2 + 1536*x^3 - 1560*x^4 - 3024*x^5 + 13824*x^6 + ...
G.f. = 1 - 48*q^3 - 216*q^6 + 1536*q^9 - 1560*q^12 - 3024*q^15 + 13824*q^18 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.

Cf. A000731, A004009, A062246, A242042.

A := Basis( ModularForms( Gamma0(9), 8/2), 30); A[1] - 48*A[2] - 216*A[3] + 1536*A[4] - 1560*A[5]; /* Michael Somos, Aug 12 2014 */

QP = QPochhammer; A = x*O[x]^40; A1 = QP[x + A]^3; A3 = QP[x^3 + A]^4; A9 = x*QP[x^9 + A]^3; s = ((A1 + 3*A9)*(A1 + 9*A9)*(A1^2 + 27*A9^2) - 48*x*A3^3 - 216*(A1*A9)^2)/A3; CoefficientList[s, x] (* Jean-François Alcover, Nov 14 2015, adapted from Michael Somos's PARI script *)
eta[q_] := q^(1/24)*QPochhammer[q]; E4[q] := 1; E4[q_] := 1 + 240 *Sum[k^3* q^k/(1 - q^k), {k, 1, 500}]; CoefficientList[Series[E4[q^3] - 48*eta[q^3]^8 - 216*(eta[q]*eta[q^9])^6/eta[q^3]^4, {q, 0, 50}], q] (* G. C. Greubel, May 01 2018 *)

{a(n) = local(A, A1, A3, A9); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A)^3; A3 = eta(x^3 + A)^4; A9 = x * eta(x^9 + A)^3; polcoeff( ((A1 + 3*A9) * (A1 + 9*A9) * (A1^2 + 27*A9^2) - 48*x*A3^3 - 216*(A1*A9)^2) / A3, n))}; /* Michael Somos, Aug 12 2014 */

Expansion of Q(q^3) - 48 * q * f(-q^3)^8 - 216 * q^2 * (f(-q) * f(-q)^9)^6 / f(-q^3)^4 in powers of q where Q(), f() are Ramanujan q-series. - Michael Somos, Aug 12 2014
Expansion of (a(q)^4 - 18 * a(q)^3*a(q^3) + 60 * a(q)^2*a(q^3)^2 - 54 * a(q)*a(q^3)^3 + 9 * a(q^3)^4) / -2 where a() is a cubic AGM theta function. - Michael Somos, Aug 12 2014
Expansion of b(q)^4 - 12 * b(q)^3*c(q^3) - 66 * b(q)^2*c(q^3)^2 - 36 * b(q)*c(q^3)^3 + 9 * c(q^3)^4 in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Aug 12 2014
Expansion of E_4(q^3) - 48 * eta(q^3)^8 - 216 * eta(q)^6 * eta(q^9)^6 / eta(q^3)^4 in powers of q. [McKay and Sebbar, equation (8.2)] - Michael Somos, Aug 12 2014
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 81 (t/i)^4 f(t) where q = exp(2 Pi i t).
a(3*n) = A004009(n) -216 * A242042(3*n).  a(3*n + 1) = -48 * A000731(n) -216 * A242042(3*n + 1).  a(3*n + 2) = -216 * A242042(3*n + 2).  - Michael Somos, Aug 12 2014

More terms from John McKay (mckay(AT)cs.concordia.ca), Apr 18 2004

More terms from Michael Somos, Aug 12 2014

A153729 Expansion of q^(-1/3) * (eta(q)^8 + 32 * eta(q^4)^8) in powers of q.

Original entry on oeis.org

1, 24, 20, 0, -70, -192, 56, 0, -125, 480, 308, 0, 110, 0, -520, 0, 57, -1680, 0, 0, 182, 1536, -880, 0, 1190, 1344, 884, 0, 0, 0, -1400, 0, -1330, -3000, 1820, 0, -646, -3840, 0, 0, -1331, 7392, 380, 0, 1120, 0, 2576, 0, 0, 2640, 1748, 0, -3850, 0, -3400, 0, 2703, -12480, -2500, 0

Offset: 0

Michael Somos, Dec 31 2008

sign

			q + 24*q^4 + 20*q^7 - 70*q^13 - 192*q^16 + 56*q^19 - 125*q^25 + ...

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

A000731(2*n) = A153728(n) = a(2*n). 24 * A000731(n) = a(4*n + 1).

QP = QPochhammer; s=QP[q]^8+32*q*QP[q^4]^8 + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015, adapted from PARI *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^8 + 32 * x * eta(x^4 + A)^8, n))}

{a(n) = local(A, p, e, x, y, a0, a1); if( n<0, 0, n = 3*n + 1; A = factor( n); prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==3, 0, if( p==2, -3 * ((e+1)%2) * (-8)^(e\2), if( p%3==2, if(e%2, 0, (-p^3) ^ (e/2)), forstep( y = sqrtint(4*p\3), sqrtint(p\3), -1, if( issquare( 4*p - 3*y^2, &x), if( x%3!=2, x = -x); break)); a0 = 1; a1 = y = x * (x^2 - 3*p); for( i=2, e, x = y*a1 - p^3*a0; a0 = a1; a1 = x); a1))))))} /* Michael Somos, Mar 01 2011 */

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2592 (t / i)^4 g(t) where q = exp(2 Pi i t) and g() is g.f. for A153728.
a(4*n + 3) = 0.
a(n) = b(3*n + 1) where b(n) is multiplicative and b(3^e) = 0^e, b(2^e) = (-3/2) * (1+(-1)^e) * (-8)^(e/2), b(p^e) = (1/2) * (1+(-1)^e) * (-p^3) ^ (e/2) if p == 2 (mod 3), b(p^e) = b(p) * b(p^(e-1)) - p^3 * b(p^(e-2)) if p == 1 (mod 3) where b(p) = x * (x^2 -3*p), 4*p = x^2 + 3*y^2, |x|<|y| and x == 2 (mod 3). - Michael Somos, Mar 01 2011

A261278 Expansion of eta(q^3)^8 + 4 * eta(q^6)^8 in powers of q.

Original entry on oeis.org

1, 4, 0, -8, 0, 0, 20, -32, 0, 0, 0, 0, -70, 80, 0, 64, 0, 0, 56, 0, 0, 0, 0, 0, -125, -280, 0, -160, 0, 0, 308, 256, 0, 0, 0, 0, 110, 224, 0, 0, 0, 0, -520, 0, 0, 0, 0, 0, 57, -500, 0, 560, 0, 0, 0, -640, 0, 0, 0, 0, 182, 1232, 0, -512, 0, 0, -880, 0, 0, 0, 0

Offset: 1

Michael Somos, Aug 14 2015

			G.f. = x + 4*x^2 - 8*x^4 + 20*x^7 - 32*x^8 - 70*x^13 + 80*x^14 + ...

G. C. Greubel, Table of n, a(n) for n = 1..2500

Cf. A000731, A153728, A261277.

A := Basis( CuspForms( Gamma0(18), 4), 72); A[1] + 4*A[2] - 8*A[4];

a[ n_] := SeriesCoefficient[ x QPochhammer[ x^3]^8 + 4 x^2 QPochhammer[ x^6]^8, {x, 0, n}];

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A)^8 + 4 * x * eta(x^6 + A)^8, n))};

A = CuspForms( Gamma0(18), 4, prec=20).basis(); A[0] + 4*A[1] - 8*A[3];

a(n) is multiplicative with a(2^(2*k)) = (-8)^k, a(2^(2*k+1)) = 4 * (-8)^k, a(3^e) = 0^e, a(p^(2*k)) = (-p)^(3^k) and a(p^(2*k+1)) = 0 if p == 5 (mod 6), a(p^e) = a(p) * a(p^(e-1)) - p^3 * a(p^(e-2)) if p == 1 (mod 6).
a(3*n) = a(6*n + 5) = 0. a(3*n + 1) = A000731(n). a(4*n) = -8 * a(n). a(6*n + 1) = A153728(n).
Convolution square of A261277.

A161969 Expansion of f(q)^8 in powers of q where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 8, 20, 0, -70, -64, 56, 0, -125, 160, 308, 0, 110, 0, -520, 0, 57, -560, 0, 0, 182, 512, -880, 0, 1190, 448, 884, 0, 0, 0, -1400, 0, -1330, -1000, 1820, 0, -646, -1280, 0, 0, -1331, 2464, 380, 0, 1120, 0, 2576, 0, 0, 880, 1748, 0, -3850, 0, -3400, 0, 2703

Offset: 0

Michael Somos, Jun 22 2009

sign

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

			G.f. = 1 + 8*x + 20*x^2 - 70*x^4 - 64*x^5 + 56*x^6 - 1258*x^8 + ...
G.f. = q + 8*q^4 + 20*q^7 - 70*q^13 - 64*q^16 + 56*q^19 - 125*q^25 + 160*q^28 + ...

Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000731, A153728.

A := Basis( CuspForms( Gamma0(36), 4), 170); A[1] + 8*A[4] + 20*A[7] - 70*A[12]; /* Michael Somos, Sep 02 2015 */

a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^8, {x, 0, n}]; (* Michael Somos, Sep 06 2015 *)

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

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, A = factor(3*n + 1); (-1)^n * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, p%3==2, if( e%2, 0, (-1)^(e/2) * p^(3*e/2)), forstep( y=sqrtint(4*p\3), sqrtint(p\3), -1, if( issquare( 4*p - 3*y^2, &x), if( x%3!=2, x=-x); break)); a0=1; a1 = y = x * (x^2 - 3*p); for( i=2, e, x = y*a1 - p^3*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Sep 06 2015 */

Expansion of q^(-1/3) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^8 in powers of q.
Euler transform of period 4 sequence [8, -16, 8, -8, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 1296 (t/i)^4 f(t) where q = exp(2 Pi i t).
a(n) = b(3*n + 1) where b() is multiplicative with b(3^e) = 0^e, b(2^e) = (1+(-1)^e)/2 * -(-8)^(e/2) if e>0, b(p^e) = (1+(-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.: Product_{k>0} (1 - (-x)^k)^8.
a(n) = (-1)^n * A000731(n).
a(4*n + 3) = a(16*n + 13) = 0. a(4*n + 1) = (-1)^n * 8 * a(n).
a(2*n) = A153728(n). - Michael Somos, Sep 06 2015

Corrected by Charles R Greathouse IV, Sep 02 2009

A258724 Expansion of f(-x)^11 / f(-x^3) + 27 * x * f(-x^3)^11 / f(-x) in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 16, 71, 0, -337, 256, -601, 0, 625, 1136, 194, 0, -529, 0, -3214, 0, 2640, -5392, 0, 0, 7199, 4096, 2903, 0, -1249, -9616, 4679, 0, 0, 0, -23927, 0, 9071, 10000, -19849, 0, 22034, 18176, 0, 0, 14641, 3104, -10942, 0, -42671, 0, 24359, 0, 0, -8464, -42121

Offset: 0

Michael Somos, Jun 08 2015

sign

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

This is a member of an infinite family of integer weight modular forms. g_1 = A033687, g_2 = A030206, g_3 = A130539, g_4 = A000731.

			G.f. = 1 + 16*x + 71*x^2 - 337*x^4 + 256*x^5 - 601*x^6 + 625*x^8 + ...
G.f. = q + 16*q^4 + 71*q^7 - 337*q^13 + 256*q^16 - 601*q^19 + ...

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. A000731, A030206, A033687, A130539.

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^11 / QPochhammer[ x^3] + 27 x QPochhammer[ x^3]^11 / QPochhammer[ x], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^11 / eta(x^3 + A) + 27 * x * eta(x^3 + A)^11 / eta(x + A), n))};

{a(n) = my(A, p, e, x, y, a0, a1); n = 3*n + 1; if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==3, 0, p%3==2, if( e%2, 0, p^(2*e)), for( x=1, sqrtint(4*p\27), if( issquare(4*p - 27*x^2, &y), break)); y = (y^2 - 2*p)^2 - 2*p^2; a0=1; a1=y; for( i=2, e, x = y*a1 - p^4*a0; a0=a1; a1=x); a1)))};

Expansion of q^(-1/3) * (eta(q)^11 / eta(q^3) + 27 * eta(q^3)^11 / eta(q)) in powers of q.
a(n) = b(3*n + 1) where b(n) is multiplicative and b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(2*e) if p == 2 (mod 3), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) if p == 1 (mod 3) where b(p) = (y^2 - 2*p)^2 - 2*p^2, 4*p = y^2 + 27*x^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27^(5/2) (t/i)^5 f(t) where q = exp(2 Pi i t).
a(4*n + 3) = 0.

A277076 Expansion of f(-x)^8 * Q(x) in powers of x where f() is a Ramanujan theta function and Q() is a Ramanujan Lambert series.

Original entry on oeis.org

1, 232, 260, -5760, 6890, 7744, 33176, -115200, 14035, 60320, 1508, 449280, -380770, -599040, 7640, 599040, -755943, 1598480, 1843200, -2620800, -988858, -2995712, 3857360, -1497600, -2004730, 7696832, 2699684, 1670400, -7188480, -11980800, 1791400, 10736640

Offset: 0

Michael Somos, Sep 27 2016

sign

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

Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

			G.f. = 1 + 232*x + 260*x^2 - 5760*x^3 + 6890*x^4 + 7744*x^5 + 33176*x^6 - 115200*x^7 + 14035*x^8 + ...
G.f. = q + 232*q^4 + 260*q^7 - 5760*q^10 + 6890*q^13 + 7744*q^16 + 33176*q^19 - 115200*q^22 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 329, 2nd equation.

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. A000731, A000739, A004009.

A := Basis( CuspForms( Gamma0(9), 8), 95); A[1] + 232*A[4];

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ QPochhammer[ x]^8 (1 + 240 Sum[ DivisorSigma[ 3, k] x^k, {k, n}]), {x, 0, n}]];
a[ n_] := SeriesCoefficient[ With[ {A1 = QPochhammer[ x]^8, A2 = QPochhammer[ x^2]^8}, A1 (A1^3 + 256 x A2^3) / (A1 A2)], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^8 * sum(k=1, n, 240 * sigma(k, 3) * x^k, 1 + A), n))};

{a(n) = my(A, A1, A2); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A)^8; A2 = eta(x^2 + A)^8; polcoeff( A1 * (A1^3 + 256 * x * A2^3) / (A1 * A2), n))};

Expansion of f(-x)^8 * (f(-x)^24 + 256 * x * f(-x^2)^24) / (f(-x) * f(-x^2))^8 in powers of x.
a(n) = b(3*n+1) where b() is multiplicative with b(p^e) = 0^e if p=3 and b(p^e) = b(p)*b(p^(e-1))  - p^7*b(p^(e-2)) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 6561 (t/i)^8 f(t) where q = exp(2 Pi i t).
Convolution of A000731 and A004009.

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