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%I #37 Sep 08 2022 08:45:18
%S 1,2,6,8,14,12,24,16,30,26,36,24,56,2,48,48,62,36,78,40,84,64,72,48,
%T 120,62,6,80,112,60,144,64,126,96,108,96,182,76,120,8,180,84,192,88,
%U 168,156,144,96,248,114,186,144,14,108,240,144,240,160,180,120
%N Theta series of quadratic form with Gram matrix [ 2, 1, 0, 1; 1, 4, 1, 0; 0, 1, 4, -2; 1, 0, -2, 8].
%C Coefficients of a theta series associated with a certain "Haupt-form" of rank 4 and level 13.
%C The Gram matrix is denoted by A in Parry 1979 on page 165.
%H W. R. Parry, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002196476">A negative result on the representation of modular forms by theta series</a>, J. Reine Angew. Math., 310 (1979), 151-170.
%F a(n) = 2 * b(n) where b() is multiplicative and b(13^e) = 1, b(p^e) = (p^(e+1) - 1) / (p - 1) otherwise. - _Michael Somos_, Mar 23 2012
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (13 t)) = 13 (t/i)^2 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Mar 23 2012
%F a(n) = 2 * A284587(n) if n>1. - _Michael Somos_, Oct 23 2019
%e G.f. = 1 + 2*q + 6*q^2 + 8*q^3 + 14*q^4 + 12*q^5 + 24*q^6 + 16*q^7 + 30*q^8 + ...
%t a[n_] := If[n == 0, 1, 2 DivisorSigma[1, n/13^IntegerExponent[n, 13]]];
%t a /@ Range[0, 59] (* _Jean-François Alcover_, Oct 23 2019, after _Michael Somos_ *)
%t a[n_] := If[n == 0, 1, 2 DivisorSum[n, Boole[!Divisible[#, 13]] # &]];
%t a /@ Range[0, 59] (* _Jean-François Alcover_, Oct 23 2019 *)
%o (PARI) {a(n) = if( n<1, n==0, 1, 2 * sigma(n / 13^valuation(n, 13)))}; /* _Michael Somos_, Mar 23 2012 */
%o (PARI) {a(n) = my(G); if( n<0, 0, G = [2, 1, 0, 1; 1, 4, 1, 0; 0, 1, 4, -2; 1, 0, -2, 8]; polcoeff( 1 + 2 * x * Ser(qfrep( G, n, 1)), n))}; /* _Michael Somos_, Mar 23 2012 */
%o (Sage) ModularForms( Gamma0(13), 2, prec=100).0; # _Michael Somos_, Jun 27 2013
%o (Magma) Basis( ModularForms( Gamma0(13), 2), 100) [1]; /* _Michael Somos_, Aug 15 2016 */
%o (Magma) [Coefficient(Basis(ModularForms(Gamma0(13), 2))[1], n) : n in [0..100] ]; // _Vincenzo Librandi_, Jun 27 2017
%Y Cf. A284587.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, May 28 2005
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