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

%I A000731 M4488 N1900 #90 Dec 29 2024 10:55:56
%S A000731 1,-8,20,0,-70,64,56,0,-125,-160,308,0,110,0,-520,0,57,560,0,0,182,
%T A000731 -512,-880,0,1190,-448,884,0,0,0,-1400,0,-1330,1000,1820,0,-646,1280,
%U A000731 0,0,-1331,-2464,380,0,1120,0,2576,0,0,-880,1748,0,-3850,0,-3400,0,2703,4160,-2500,0,3458
%N A000731 Expansion of Product (1 - x^k)^8 in powers of x.
%C A000731 Number 22 of the 74 eta-quotients listed in Table I of Martin (1996).
%C A000731 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.
%C A000731 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
%C A000731 a(n)=0 if and only if A033687(n)=0 (see the Han-Ono paper). - _Emeric Deutsch_, May 16 2008
%C A000731 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D A000731 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.
%D A000731 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000731 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000731 Seiichi Manyama, <a href="/A000731/b000731.txt">Table of n, a(n) for n = 0..10000</a>
%H A000731 Matthew Boylan, <a href="http://dx.doi.org/10.1016/S0022-314X(02)00037-9">Exceptional congruences for the coefficients of certain eta-product newforms</a>, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
%H A000731 Shaun Cooper, Minchael D. Hirschhorn and Richard Lewis, <a href="https://doi.org/10.1023/A:1009827103485">Powers of Euler's Product and Related Identities</a>, The Ramanujan Journal, Vol. 4 (2), 137-155 (2000).
%H A000731 Slawomir Cynk and Klaus Hulek, <a href="http://arXiv.org/abs/math/0509424">Construction and examples of higher-dimensional modular Calabi-Yau manifolds</a>, arXiv:math/0509424 [math.AG], 2005-2006.
%H A000731 Steven R. Finch, <a href="https://arxiv.org/abs/math/0701251">Powers of Euler's q-Series</a>, arXiv:math/0701251 [math.NT], 2007.
%H A000731 Guo-Niu Han and Ken Ono, <a href="https://doi.org/10.1007/s00026-011-0096-3">Hook Lengths and 3-Cores</a>, Ann. Comb. (2011) Vol. 15, 305-312. See also <a href="https://arxiv.org/abs/0805.2461">arXiv:0805.2461</a>, [math.NT], 2008.
%H A000731 Iva Kodrnja and Helena Koncul, <a href="https://hrcak.srce.hr/file/470068">Polynomials vanishing on a basis of S_m(Gamma_0(N))</a>, Glasnik Matematički (2024) Vol. 59, No. 79, 313-325. See p. 319.
%H A000731 Yves Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H A000731 Morris Newman, <a href="/A000727/a000727.pdf">A table of the coefficients of the powers of eta(tau)</a>, Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
%H A000731 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H A000731 <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%F A000731 Expansion of q^(-1/3) * eta(q)^8 in powers of q.
%F A000731 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
%F A000731 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
%F A000731 Euler transform of period 1 sequence [ -8, ...].
%F A000731 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
%F A000731 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
%F A000731 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
%F A000731 G.f.: Product_{k>0} (1 - x^k)^8.
%F A000731 a(2*n) = A153728(n). - _Michael Somos_, Sep 29 2011
%F A000731 a(4*n + 1) = -8 * a(n). - _Michael Somos_, Dec 06 2004
%F A000731 a(4*n + 3) = a(16*n + 13) = 0. - _Michael Somos_, Oct 19 2005
%F A000731 A092342(n) = a(n) + 81*A033690(n-1). - _Michael Somos_, Aug 22 2007
%F A000731 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
%F A000731 a(0) = 1, a(n) = -(8/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Mar 27 2017
%F A000731 G.f.: exp(-8*Sum_{k>=1} x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 05 2018
%F A000731 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
%F A000731 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
%e A000731 G.f. = 1 - 8*x + 20*x^2 - 70*x^3 + 64*x^4 + 56*x^5 - 125*x^6 - 160*x^7 + ...
%e A000731 G.f. = q - 8*q^4 + 20*q^7 - 70*q^13 + 64*q^16 + 56*q^19 - 125*q^25 - ...
%t A000731 a[ n_] := SeriesCoefficient[ QPochhammer[ x]^8, {x, 0, n}]; (* _Michael Somos_, Sep 29 2011 *)
%t A000731 a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}]^8, {x, 0, n}]; (* _Michael Somos_, Dec 09 2013 *)
%o A000731 (PARI) {a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n))^8, n))};
%o A000731 (PARI) {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 */
%o A000731 (Sage) CuspForms( Gamma0(9), 4, prec=56).0; # _Michael Somos_, May 28 2013
%o A000731 (Magma) Basis( CuspForms( Gamma0(9), 4), 56) [1]; /* _Michael Somos_, Dec 09 2013 */
%Y A000731 Cf. A033687, A033690, A092342, A153728.
%Y A000731 Powers of Euler's product: A000594, A000727 - A000731, A000735, A000739, A002107, A010815 - A010840.
%K A000731 sign,easy
%O A000731 0,2
%A A000731 _N. J. A. Sloane_
%E A000731 Corrected by _Charles R Greathouse IV_, Sep 02 2009