A033690 -id:A033690 - cp's OEIS frontend

A000731 Expansion of Product (1 - x^k)^8 in powers of x.

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, 4160, -2500, 0, 3458

Offset: 0

N. J. A. Sloane

Number 22 of the 74 eta-quotients listed in Table I of Martin (1996).
Denoted by g_4(q) in Cynk and Hulek in Remark 3.4 on page 12 as the unique level 9 form of weight 4.
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. - Michael Somos, Aug 24 2012
a(n)=0 if and only if A033687(n)=0 (see the Han-Ono paper). - Emeric Deutsch, May 16 2008
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 8*x + 20*x^2 - 70*x^3 + 64*x^4 + 56*x^5 - 125*x^6 - 160*x^7 + ...
G.f. = q - 8*q^4 + 20*q^7 - 70*q^13 + 64*q^16 + 56*q^19 - 125*q^25 - ...

Newman, Morris; A table of the coefficients of the powers of eta(tau). Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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
Matthew Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
Shaun Cooper, Minchael D. Hirschhorn and Richard Lewis, Powers of Euler's Product and Related Identities, The Ramanujan Journal, Vol. 4 (2), 137-155 (2000).
Slawomir Cynk and Klaus Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Guo-Niu Han and Ken Ono, Hook Lengths and 3-Cores, Ann. Comb. (2011) Vol. 15, 305-312. See also arXiv:0805.2461, [math.NT], 2008.
Iva Kodrnja and Helena Koncul, Polynomials vanishing on a basis of S_m(Gamma_0(N)), Glasnik Matematički (2024) Vol. 59, No. 79, 313-325. See p. 319.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Morris 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]
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. A033687, A033690, A092342, A153728.

Powers of Euler's product: A000594, A000727 - A000731, A000735, A000739, A002107, A010815 - A010840.

Basis( CuspForms( Gamma0(9), 4), 56) [1]; /* Michael Somos, Dec 09 2013 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^8, {x, 0, n}]; (* Michael Somos, Sep 29 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}]^8, {x, 0, n}]; (* Michael Somos, Dec 09 2013 *)

{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, n = 3*n + 1; A = factor(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, Aug 23 2006 */

CuspForms( Gamma0(9), 4, prec=56).0; # Michael Somos, May 28 2013

Expansion of q^(-1/3) * eta(q)^8 in powers of q.
Expansion of q^(-1/3) * b(q)^3 * c(q) / 3 in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Nov 08 2006
Expansion of q^(-1) * b(q) * c(q)^3 / 27 in powers of q^3 where b(), c() are cubic AGM theta functions. - Michael Somos, Nov 08 2006
Euler transform of period 1 sequence [ -8, ...].
a(n) = b(3*n + 1) where b(n) is multiplicative and b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * (-1)^(e/2) * p^(3*e/2) if p == 2 (mod 3), b(p^e) = b(p)*b(p^(e-1)) - b(p^(e-2))*p^3 if p == 1 (mod 3) where b(p) = (x^2 - 3*p)*x, 4*p = x^2 + 3*y^2, |x|<|y| and x == 2 (mod 3). - Michael Somos, Aug 23 2006
Given g.f. A(x), then B(x) = x * A(x^3) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v^3 - u * w * (u + 16 * w). - Michael Somos, Feb 19 2007
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). - Michael Somos, Sep 29 2011
G.f.: Product_{k>0} (1 - x^k)^8.
a(2*n) = A153728(n). - Michael Somos, Sep 29 2011
a(4*n + 1) = -8 * a(n). - Michael Somos, Dec 06 2004
a(4*n + 3) = a(16*n + 13) = 0. - Michael Somos, Oct 19 2005
A092342(n) = a(n) + 81*A033690(n-1). - Michael Somos, Aug 22 2007
Sum_{n>=0} a(n) * q^(3*n + 1) = (Sum_{i,j,k in Z} (i-j) * (j-k) * (k-i) * q^((i*i + j*j + k*k) / 2)) / 2 where 0 = i+j+k, i == 1 (mod 3), j == 2 (mod 3), and k == 0 (mod 3). - Michael Somos, Sep 22 2014
a(0) = 1, a(n) = -(8/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017
G.f.: exp(-8*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
Let M = p_1*...*p_k be a positive integer whose prime factors p_i (not necessarily distinct) are all congruent to 2 (mod 3). Then a( M^2*n + (M^2 - 1)/3 ) = (-1)^k*M^3*a(n). See Cooper et al., Theorem 1. - Peter Bala, Dec 01 2020
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^3)^(e/2) if p == 2 (mod 3), b(p^e) = (((x+sqrt(-3)*y)/2)^(3*e+3) - ((x-sqrt(-3)*y)/2)^(3*e+3))/(((x+sqrt(-3)*y)/2)^3 - ((x-sqrt(-3)*y)/2)^3) if p == 1 (mod 3) where 4*p = x^2 + 3*y^2, |x|<|y| and x == 2 (mod 3). - Jianing Song, Mar 19 2022

Corrected by Charles R Greathouse IV, Sep 02 2009

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

A133078 Expansion of s(q)^4 in powers of q where s() is a cubic AGM function.

Original entry on oeis.org

1, -12, 54, -84, -147, 756, -756, -1212, 3510, -2028, -3402, 7992, -6132, -5964, 18576, -10584, -14619, 29484, -18252, -21084, 55188, -28896, -35964, 73008, -49140, -46128, 118692, -54516, -73896, 146340, -95256, -92148, 224694, -111888, -132678, 260064

Offset: 0

Michael Somos, Sep 08 2007

sign

Cubic AGM theta functions: r(q) (see A004016), s(q) (A005928), t(q) (A005882).

			G.f. = 1 - 12*q + 54*q^2 - 84*q^3 - 147*q^4 + 756*q^5 - 756*q^6 - 1212*q^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)

Cf. A033690.

QP = QPochhammer; A133078[n_] := SeriesCoefficient[(QP[q]^3/QP[q^3])^4, {q, 0, n}]; Table[A133078[n], {n, 0, 50}] (* G. C. Greubel, Oct 20 2017 *)

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

Expansion of (eta(q)^3 / eta(q^3))^4 in powers of q.
Euler transform of period 3 sequence [ -12, -12, -8, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 729 (t/i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033690.
G.f.: ( Product_{k>0} (1 - x^k)^3 / (1 - x^(3*k)) )^4.

A242042 Expansion of (b(q) * c(q^3) / 3)^2 in powers of q where b(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, -6, 9, 14, -54, 36, 65, -162, 126, 148, -438, 252, 344, -756, 513, 546, -1458, 756, 1022, -2064, 1332, 1352, -3510, 1764, 2198, -4374, 2808, 2710, -6804, 3276, 4161, -7992, 4914, 4816, -11826, 5616, 6860, -13188, 8190, 7658, -18576, 8892, 10804, -20412

Offset: 2

Michael Somos, Aug 12 2014

sign

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

In McKay and Sebbar on page 274 in equation (8.2) the last term on the right side is a multiple of the g.f.

			G.f. = q^2 - 6*q^3 + 9*q^4 + 14*q^5 - 54*q^6 + 36*q^7 + 65*q^8 - 162*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 2..2500
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann. 318 (2000), no. 2, 255-275. MR1795562 (2001m:11063)

Cf. A033680, A106401, A198956.

A := Basis( ModularForms( Gamma0(9), 4), 19); A[3] - 6*A[4] + 9*A[5];

a[ n_] := SeriesCoefficient[ q^2 (QPochhammer[ q] QPochhammer[ q^9])^6 / QPochhammer[ q^3]^4, {q, 0, n}];

{a(n) = my(A); if( n<2, 0, n-=2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^9 + A))^6 / eta(x^3 + A)^4, n))};

Expansion of (eta(q) * eta(q^9))^6 / eta(q^3)^4 in powers of q.
Euler transform of period 9 sequence [ -6, -6, -2, -6, -6, -2, -6, -6, -8, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 81 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: x^2 * Product_{k>0} (1 - x^k)^6 * (1 - x^(9*k))^6 / (1 - x^(3*k))^4.
Convolution square of A106401.
a(3*n) = -6 * A198956(n). a(3*n + 1) = 9 * A033690(n).

cp's OEIS Frontend

A000731 Expansion of Product (1 - x^k)^8 in powers of x.

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

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A133078 Expansion of s(q)^4 in powers of q where s() is a cubic AGM function.

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A242042 Expansion of (b(q) * c(q^3) / 3)^2 in powers of q where b(), c() are cubic AGM theta functions.

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A023920 Theta series of A*_8 lattice.

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