A153728 -id:A153728 - 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

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

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

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

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

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

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