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A002408 Expansion of 8-dimensional cusp form.

%I A002408 #59 May 13 2025 14:58:54
%S A002408 0,1,-8,28,-64,126,-224,344,-512,757,-1008,1332,-1792,2198,-2752,3528,
%T A002408 -4096,4914,-6056,6860,-8064,9632,-10656,12168,-14336,15751,-17584,
%U A002408 20440,-22016,24390,-28224,29792,-32768,37296,-39312,43344,-48448,50654,-54880,61544,-64512,68922
%N A002408 Expansion of 8-dimensional cusp form.
%C A002408 Essentially the same as A007331.
%C A002408 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A002408 "For Gamma, it is known that any modular form is a weighted homogeneous polynomial in Theta_Z, which has weight 1/2, and the modular form delta_8(t) := e^(Pi i tau) Product_{m=1..oo} ((1 - e^(Pi i m tau)) (1 + e^(2 Pi i m tau)))^8 = e^(Pi i tau) - 8 e^(2 Pi i tau) + 28 e^(3 Pi i tau) - 64 e^(4 Pi i tau) + 126 e^(5 Pi i tau) ... of weight 4." [Elkies, p. 1242]
%D A002408 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 187.
%D A002408 Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and Modular Forms, Vieweg 1994, p. 133.
%H A002408 T. D. Noe, <a href="/A002408/b002408.txt">Table of n, a(n) for n = 0..1000</a>
%H A002408 Noam D. Elkies, <a href="https://www.ams.org/notices/200010/fea-elkies-1.pdf">Lattices, Linear Codes and Invariants, Part I</a>, Notices of the Amer. Math. Soc., 47 (No. 10, Nov. 2000), 1238-1245, see p. 1242.
%H A002408 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H A002408 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A002408 Expansion of (eta(q)* eta(q^4) / eta(q^2))^8 in powers of q. - _Michael Somos_, Jul 16 2004
%F A002408 Euler transform of period 4 sequence [-8, 0, -8, -8, ...]. - _Michael Somos_, Jul 16 2004
%F A002408 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = +u^4*w*v + 16*u^3*w*v^2 + 16*u^2*w^2*v^2 + 256*u^3*w^3 + 256*u^3*w^2*v + 4096*u^2*w^3*v + 4096*u*w^4*v + 4096*u*w^3*v^2 - u^2*v^4 - 16*u^2*w*v^3 - 256*u*w^2*v^3 - 256*w^2*v^4. - _Michael Somos_, May 31 2005
%F A002408 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^4*u6^4 + u1^3*u2*u3^3*u6 + 2*u1*u2^3*u3*u6^3 - u2^4*u3^4.
%F A002408 Expansion of q * psi(-q)^8 in powers of q where psi() is a Ramanujan theta function. - _Michael Somos_, Mar 20 2008
%F A002408 a(n) is multiplicative with a(2^e) = -8^e if e>0, a(p^e) = ((p^3)^(e+1) - 1) / (p^3 - 1). - _Michael Somos_, Mar 20 2008
%F A002408 G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 16 (t/i)^4 f(t) where q = exp(2 Pi i t).
%F A002408 G.f.: x * (Product_{k>0} (1 - x^(2*k-1)) * (1 - x^(4*k)))^8.
%F A002408 a(n) = -(-1)^n * A007331(n).
%F A002408 a(2*n) = -8 * A007331(n). a(2*n + 1) = A045823(n). - _Michael Somos_, May 25 2014
%F A002408 Dirichlet g.f.: zeta(s-3) * zeta(s) * (1 - 1/2^s) * (1 - 1/2^(s-4)). - _Amiram Eldar_, Nov 03 2023
%e A002408 G.f. = q - 8*q^2 + 28*q^3 - 64*q^4 + 126*q^5 - 224*q^6 + 344*q^7 ...
%p A002408 q*product((1-q^(2*k-1))^8*(1-q^(4*k))^8, k=1..75);
%t A002408 a[0] = 0; a[n_] := -(-1)^n*Sum[ Mod[n/d, 2]*d^3, {d, Divisors[n]}]; Table[a[n], {n, 0, 41}] (* _Jean-François Alcover_, Jan 27 2012, after _Michael Somos_ *)
%t A002408 a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^4] / QPochhammer[ q^2])^8, {q, 0, n}]; (* _Michael Somos_, May 25 2014 *)
%o A002408 (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) / eta(x^2 + A))^8, n))}; /* _Michael Somos_, Jul 16 2004 */
%o A002408 (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (prod(k=1, n, (1 -( k%4==0) * x^k) * (1 - (k%2==1) * x^k), 1 + A))^8, n))}; /* _Michael Somos_, Jul 16 2004 */
%o A002408 (PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, (n/d%2) * d^3))}; /* _Michael Somos_, May 31 2005 */
%o A002408 (Sage) A = ModularForms( Gamma0(4), 4, prec=70) . basis(); A[1] - 8*A[2] # _Michael Somos, May 25 2014
%o A002408 (Python)
%o A002408 from sympy import divisors
%o A002408 def a(n): return 0 if n == 0 else -(-1)**n * sum([((n//d)%2) * d**3 for d in divisors(n)])
%o A002408 print([a(n) for n in range(101)])  # _Indranil Ghosh_, Jun 24 2017
%Y A002408 Cf. A000122, A000700, A007331, A010054, A045823, A121373, .
%K A002408 sign,nice,easy,mult
%O A002408 0,3
%A A002408 _N. J. A. Sloane_ and _Mira Bernstein_