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A005883 Theta series of square lattice with respect to deep hole.

%I A005883 M3319 #34 Feb 16 2025 08:32:29
%S A005883 4,8,4,8,8,0,12,8,0,8,8,8,4,8,0,8,16,0,8,0,4,16,8,0,8,8,0,8,8,8,4,16,
%T A005883 0,0,8,0,16,8,8,8,0,0,12,8,0,8,16,0,8,8,0,16,0,0,0,16,12,8,8,0,8,8,0,
%U A005883 0,8,8,16,8,0,8,8,0,12,8,0,0,16,0,8,8,0,24,0,8,8,0,0,8,8,0,4,16,8,8,16,0,0
%N A005883 Theta series of square lattice with respect to deep hole.
%C A005883 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A005883 In [Jacobi 1829] on page 105 is equation 18: "2 k K / Pi = 4 sqrt(q) + 8 sqrt(q^5) + 4 sqrt(q^9) [...]". - _Michael Somos_, Sep 09 2012
%D A005883 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
%D A005883 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A005883 C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, 1829.
%H A005883 G. C. Greubel, <a href="/A005883/b005883.txt">Table of n, a(n) for n = 0..1000</a>
%H A005883 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A005883 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A005883 Expansion of Jacobi theta constant q^(-1/2) * theta_2(q)^2 in powers of q^2. - _Michael Somos_, Oct 31 2006
%F A005883 G.f.: 4 * (Product_{k>0} (1 - x^k) * (1 + x^(2*k))^2)^2. - _Michael Somos_, Oct 31 2006
%F A005883 From _Michael Somos_, Sep 09 2012: (Start)
%F A005883 Expansion of 4 * psi(x)^2 in powers of x where psi() is a Ramanujan theta function.
%F A005883 Expansion of q^(-1) * (1/2) * (1 - k') * K / (Pi/2) in powers of q^4 where k', K are Jacobi elliptic functions.
%F A005883 Expansion of q^(-1/2) * k * K / (Pi/2) in powers of q^2 where k, K are Jacobi elliptic functions.
%F A005883 Expansion of q^(-1/4) * 2 * k^(1/2) * K / (Pi/2) in powers of q where k, K are Jacobi elliptic functions.
%F A005883 Expansion of 4 * q^(-1/4) * eta(q^2)^4 / eta(q)^2 in powers of q.
%F A005883 a(n) = 4 * A008441(n). (End)
%e A005883 4 + 8*x + 4*x^2 + 8*x^3 + 8*x^4 + 12*x^6 + 8*x^7 + 8*x^9 + 8*x^10 + 8*x^11 + ...
%e A005883 4*q + 8*q^3 + 4*q^5 + 8*q^7 + 8*q^9 + 12*q^13 + 8*q^15 + 8*q^19 + 8*q^21 + ...
%e A005883 Theta = 4*q^(1/2) + 8*q^(5/2) + 4*q^(9/2) + 8*q^(13/2) + 8*q^(17/2) + ...
%t A005883 a[0] = 4; a[n_] := 4*DivisorSum[4n+1, (-1)^Quotient[#, 2]&]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Nov 04 2015, translated from PARI *)
%t A005883 s = 4*(QPochhammer[q^2]^4/QPochhammer[q]^2)+O[q]^100; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 09 2015 *)
%o A005883 (PARI) {a(n) = if( n<0, 0, n = 4*n + 1; 4 * sumdiv(n, d, (-1)^(d\2)))} /* _Michael Somos_, Oct 31 2006 */
%Y A005883 Cf. A008441.
%K A005883 nonn
%O A005883 0,1
%A A005883 _N. J. A. Sloane_