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A033718 Product theta3(q^d); d | 5.

%I A033718 #45 Feb 16 2025 08:32:36
%S A033718 1,2,0,0,2,2,4,0,0,6,0,0,0,0,4,0,2,0,0,0,2,8,0,0,4,2,0,0,0,4,4,0,0,0,
%T A033718 0,0,6,0,0,0,0,4,0,0,0,6,4,0,0,6,0,0,0,0,8,0,4,0,0,0,0,4,0,0,2,0,0,0,
%U A033718 0,8,4,0,0,0,0,0,0,0,0,0,2,10,0,0,8,0,4,0,0,4,0,0,0,0,4,0,4,0,0,0,2,4,0,0,0,8,0,0,0,4,0,0,0,0,0,0,4,0,0,0,4,2,0,0,0,2,12
%N A033718 Product theta3(q^d); d | 5.
%C A033718 Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k>=0} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).
%C A033718 Number of representations of n as a sum of five times a square and a square. - _Ralf Stephan_, May 14 2007
%D A033718 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.
%H A033718 Robert Israel, <a href="/A033718/b033718.txt">Table of n, a(n) for n = 0..10000</a>
%H A033718 A. Berkovich and H. Yesilyurt, <a href="http://arXiv.org/abs/math.NT/0611300">Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms</a>, arXiv:math/0611300 [math.NT], 2006-2007.
%H A033718 N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%H A033718 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A033718 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A033718 Theta series of lattice with Gram matrix [1 0 / 0 5].
%F A033718 Expansion of phi(q)phi(q^5) in powers of q where phi(q) is a Ramanujan theta function.
%F A033718 Euler transform of period 20 sequence [ 2, -3, 2, -1, 4, -3, 2, -1, 2, -6, 2, -1, 2, -3, 4, -1, 2, -3, 2, -2, ...]. - _Michael Somos_, Aug 13 2006
%F A033718 If p is prime then a(p) is nonzero iff p is in A033205.
%F A033718 0=a(n)a(2n) and 2*A035170(n) = a(n) + a(2n) if n>0. - _Michael Somos_, Oct 21 2006
%F A033718 a(n) is nonzero iff n is in A020669. - _Robert Israel_, Dec 22 2015
%F A033718 a(0) = 1, a(n) = (1+(-1)^t)b(n) for n > 0, where t is the number of prime factors of n, counting multiplicity, which are == 2,3,7 (mod 20), and b() is multiplicative with b(p^e) = (e+1) for primes p == 1,3,7,9 (mod 20) and b(p^e) = (1+(-1)^e)/2 for primes p == 11,13,17,19 (mod 20). (This formula is Corollary 3.3 in the Berkovich-Yesilyurt paper) - _Jeremy Lovejoy_, Nov 12 2024
%p A033718 S:= series(JacobiTheta3(0,q)*JacobiTheta3(0,q^5), q, 1001):
%p A033718 seq(coeff(S,q,j),j=0..1000); # _Robert Israel_, Dec 22 2015
%t A033718 terms = 127; s = EllipticTheta[3, 0, q] EllipticTheta[3, 0, q^5] + O[q]^terms; CoefficientList[s, q] (* _Jean-François Alcover_, Jul 04 2017 *)
%o A033718 (PARI) {a(n)=if(n<1, n==0, qfrep([1,0;0,5],n)[n]*2)} /* _Michael Somos_, Aug 13 2006 */
%o A033718 (PARI)
%o A033718 N=666;  x='x+O('x^N);
%o A033718 T3(x)=1+2*sum(n=1,ceil(sqrt(N)),x^(n*n));
%o A033718 Vec(T3(x)*T3(x^5))
%o A033718 /* _Joerg Arndt_, Sep 21 2012 */
%Y A033718 Cf. A000700, A000122, A010054, A020669, A033205, A121373.
%K A033718 nonn
%O A033718 0,2
%A A033718 _N. J. A. Sloane_