This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A271231 #51 Dec 12 2023 18:48:49
%S A271231 0,1,0,1,0,-2,0,0,0,1,0,-4,0,-2,0,-2,0,2,0,4,0,0,0,8,0,-1,0,1,0,6,0,
%T A271231 -8,0,-4,0,0,0,6,0,-2,0,-6,0,-4,0,-2,0,0,0,-7,0,2,0,-2,0,8,0,4,0,-4,0,
%U A271231 -2,0,0,0,4,0,4,0,8,0,-8,0,10,0,-1,0,0,0,8,0,1,0,4,0,-4,0,6,0,-6,0,0,0,-8,0,-8,0,2,0,-4,0,-18,0,-16
%N A271231 Expansion of the modular cusp form ( eta(q^4) * eta(q^12) )^4 / ( eta(q^2) * eta(q^6) * eta(q^8) * eta(q^24) ), where eta is Dedekind's eta function.
%C A271231 The modularity pattern of the elliptic curve y^2 = x^3 + x^2 + x considered modulo prime(m) is seen from a(prime(m)) = prime(m) - N(prime(m)) = A271230(m), where N(prime(m))= A271229(m) is the number of solutions of this congruence. That is, the p-defect coincides with the prime indexed expansion coefficient (here for all primes).
%C A271231 This modular cusp form of weight 2 and level N = 48 = 2^4*3 is Nr. 54 in Martin's Table 1 (corrected by giving the 24 the missing exponent -1). See also the Michael Somos link where this correction has been observed.
%C A271231 This modular cusp form is a simultaneous eigenform of every Hecke operators T_p, with p a prime not 2 or 3 (bad primes) with eigenvalue lambda(p) = a(p). (See the Martin reference, Proposition 33, p. 4851.)
%C A271231 In the Martin and Ono reference, p. 3173 (Theorem 2), this cusp form appears (in the corrected version) in the row Conductor 48, and it is there related to the elliptic curve y^2 = x^3 + x^2 - 4*x - 4. The p-defects of this curve coincide with the ones of the curve y^2 = x^3 + x^2 + x modulo primes p given in A271230. - _Wolfdieter Lang_, Apr 21 2016
%C A271231 Multiplicative. See A159819 for formula. - _Andrew Howroyd_, Aug 06 2018
%H A271231 Seiichi Manyama, <a href="/A271231/b271231.txt">Table of n, a(n) for n = 0..10000</a>
%H A271231 Y. 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 A271231 Yves Martin and Ken Ono, <a href="http://dx.doi.org/10.1090/S0002-9939-97-03928-2">Eta-Quotients and Elliptic Curves</a>, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.
%H A271231 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%F A271231 a(2*n+1) = A159819(n), a(2*n) = 0.
%F A271231 O.g.f.: Expansion in q = exp(2*Pi*i*z) with Im(z) > 0 of (eta(4*z)*eta(12*z))^4 / (eta(2*z)*eta(6*z)*eta(8*z)*eta(24*z)), where eta(z) = q^(1/24)*Product_{n >= 1} (1 - q^n) is the Dedekind function with q = q(z) given above, and i is the imaginary unit.
%F A271231 a(prime(m)) = A271230(m), m >= 1.
%e A271231 n=2: a(2) = A271230(1) = 0.
%e A271231 n=5: a(5) = A271230(3) = -2.
%e A271231 See the example section of A271229 for the solutions for the first primes.
%t A271231 QP = QPochhammer;
%t A271231 a[n_] := If[OddQ[n], SeriesCoefficient[QP[-x] QP[x^2] QP[-x^3] QP[x^6], {x, 0, (n-1)/2}], 0];
%t A271231 a /@ Range[0, 100] (* _Jean-François Alcover_, Sep 19 2019 *)
%o A271231 (PARI) q='q+O('q^220); concat([0], Vec( (eta(q^4)*eta(q^12))^4 / (eta(q^2)*eta(q^6)*eta(q^8)*eta(q^24) ) ) ) \\ _Joerg Arndt_, Sep 12 2016
%Y A271231 Cf. A159819, A271229, A271230.
%K A271231 sign,easy,mult
%O A271231 0,6
%A A271231 _Wolfdieter Lang_, Apr 19 2016