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%I M5270 #30 Mar 12 2021 22:24:41
%S 1,40,90,240,200,560,400,800,730,1240,752,1840,1200,2000,1600,2720,
%T 1480,3680,2250,3280,2800,4320,2800,5920,2960,5240,3760,6720,4000,
%U 7920,4800,6720,5850,8960,4320,10720,6200,9840,7600,11040,5872,12960,7520,12400
%N Theta series of D_5 lattice.
%C Ramanujan theta functions: f(q) := Prod_{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..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 118.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H John Cannon, <a href="/A005930/b005930.txt">Table of n, a(n) for n = 0..5000</a>
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D5.html">Home page for this lattice</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F G.f.: (theta_3(q^(1/2))^5+theta_4(q^(1/2))^5)/2
%F Expansion of ( phi(q)^5 + phi(-q)^5 ) / 2 in powers of q^2 where phi() is a Ramanujan theta function. - _Michael Somos_, Sep 14 2007
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 64 2^(1/2) (t/i)^(5/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A008422.
%e 1 + 40*q^2 + 90*q^4 + 240*q^6 + 200*q^8 + 560*q^10 + 400*q^12 + 800*q^14 + ...
%t terms = 44; phi[q_] := EllipticTheta[3, 0, q]; s = (phi[q]^5 + phi[-q]^5)/2 + O[q]^(2 terms); DeleteCases[CoefficientList[s, q], 0][[1 ;; terms]] (* _Jean-François Alcover_, Jul 04 2017, after _Michael Somos_ *)
%o (PARI) {a(n)=if(n<0, 0, n*=2; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n))^5, n))} /* _Michael Somos_, Nov 03 2006 */
%Y A000132(2n) = a(n). A008422 gives dual lattice.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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