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 A005879 Theta series of D_4 lattice with respect to deep hole. (Formerly M4509) 4
 8, 32, 48, 64, 104, 96, 112, 192, 144, 160, 256, 192, 248, 320, 240, 256, 384, 384, 304, 448, 336, 352, 624, 384, 456, 576, 432, 576, 640, 480, 496, 832, 672, 544, 768, 576, 592, 992, 768, 640, 968, 672, 864, 960, 720, 896, 1024, 960, 784, 1248, 816, 832, 1536 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). The D_4 lattice is the set of all integer quadruples [a, b, c, d] where a + b + c + d is even. The deep holes are quadruples [a, b, c, d] where each coordinate is half an odd integer and where a + b + c + d is even. - Michael Somos, May 23 2102 REFERENCES J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 118. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..10000 G. Nebe and N. J. A. Sloane, Home page for this lattice Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of Jacobi theta_2(q)^4/(2q) in powers of q^2. - Michael Somos, Apr 11 2004 Expansion of q^(-1/2) * 8 * (eta(q^2)^2 / eta(q))^4 in powers of q. - Michael Somos, Apr 11 2004 Expansion of 8 * psi(x)^4 in powers of x where psi() is a Ramanujan theta function. - Michael Somos, May 23 2012 Expansion of (phi(q)^4 - phi(-q)^4) / (2 * q) in powers of q^2. - Michael Somos, May 23 2012 G.f.: 8 * (Product_{k>0} (1 - x^k) * (1 + x^k)^2)^4. - Michael Somos, Apr 11 2004 a(n) = 8 * A008438(n) = 4 * A005880(n) = A000118(2*n + 1) = - A096727(2*n + 1). - Michael Somos, Nov 01 2006 EXAMPLE 8 + 32*x + 48*x^2 + 64*x^3 + 104*x^4 + 96*x^5 + 112*x^6 + 192*x^7 + ... 8*q + 32*q^3 + 48*q^5 + 64*q^7 + 104*q^9 + 96*q^11 + 112*q^13 + ... . For n = 2 the objects counted are the ways to represent the integer 5 = (2*n+1) as a sum of 4 squares, 0 and negative numbers allowed. [-2,-1,0,0], [-2,0,-1,0], [-2,0,0,-1], [-2,0,0,1], [-2,0,1,0], [-2,1,0,0], [-1,-2,0,0], [-1,0,-2,0], [-1,0,0,-2], [-1,0,0,2], [-1,0,2,0], [-1,2,0,0], [0,-2,-1,0], [0,-2,0,-1], [0,-2,0,1], [0,-2,1,0], [0,-1,-2,0], [0,-1,0,-2], [0,-1,0,2], [0,-1,2,0], [0,0,-2,-1], [0,0,-2,1], [0,0,-1,-2], [0,0,-1,2], [0,0,1,-2], [0,0,1,2], [0,0,2,-1], [0,0,2,1], [0,1,-2,0], [0,1,0,-2], [0,1,0,2], [0,1,2,0], [0,2,-1,0], [0,2,0,-1], [0,2,0,1], [0,2,1,0], [1,-2,0,0], [1,0,-2,0], [1,0,0,-2], [1,0,0,2], [1,0,2,0], [1,2,0,0], [2,-1,0,0], [2,0,-1,0], [2,0,0,-1], [2,0,0,1], [2,0,1,0], [2,1,0,0]. - Peter Luschny, Nov 03 2015 MAPLE S:= series(JacobiTheta2(0, q)^4/(2*q), q, 202): seq(coeff(S, q, 2*j), j=0..100); # Robert Israel, Nov 03 2015 MATHEMATICA (* a(n) gives the number of ways to represent the integer 2n+1 as a sum of 4 squares *) a[n_] := SquaresR[4, 2n+1]; Table[a[n], {n, 0, 52}] (* Jean-François Alcover, Nov 03 2015 *) terms = 53; QP = QPochhammer; s = 8 QP[q^2]^8/QP[q]^4 + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 07 2017, after Michael Somos *) PROG (PARI) {a(n) = if( n<0, 0, 8 * sigma(2*n + 1))} /* Michael Somos, Apr 11 2004 */ (PARI) q='q+O('q^66); Vec(8*(eta(q^2)^2/eta(q))^4) \\ Joerg Arndt, Nov 03 2015 CROSSREFS Cf. A000118, A005880, A008438, A096727. Sequence in context: A144096 A127988 A129749 * A067519 A253295 A290960 Adjacent sequences:  A005876 A005877 A005878 * A005880 A005881 A005882 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified December 11 12:52 EST 2018. Contains 318049 sequences. (Running on oeis4.)