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 A004020 Theta series of square lattice with respect to edge. (Formerly M0931) 8
 2, 4, 2, 4, 4, 0, 6, 4, 0, 4, 4, 4, 2, 4, 0, 4, 8, 0, 4, 0, 2, 8, 4, 0, 4, 4, 0, 4, 4, 4, 2, 8, 0, 0, 4, 0, 8, 4, 4, 4, 0, 0, 6, 4, 0, 4, 8, 0, 4, 4, 0, 8, 0, 0, 0, 8, 6, 4, 4, 0, 4, 4, 0, 0, 4, 4, 8, 4, 0, 4, 4, 0, 6, 4, 0, 0, 8, 0, 4, 4, 0, 12, 0, 4, 4, 0, 0, 4, 4, 0, 2, 8, 4, 4, 8, 0, 0, 4, 0, 4, 4, 4, 4, 0 (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). Number of solutions in integers of n = x^2 + y^2 + y. REFERENCES J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA G.f.: 2 * (Sum_{k>0} x^((k^2 - k)/2))^2 = (Sum_{k in Z} x^(k^2 + k)) * (Sum_{k in Z} x^(k^2)). Expansion of q^(-1/2) * c(q) / 2 in powers of q^2 where c(q) is the third function in the quadratic Gauss AGM. - Michael Somos, Feb 10 2006 Expansion of 2 * phi(x) * psi(x^2) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 10 2006 a(n) = 2*A008441(n) = A004531(4*n + 1). EXAMPLE G.f. = 2 + 4*x + 2*x^2 + 4*x^3 + 4*x^4 + 6*x^6 + 4*x^7 + 4*x^9 + 4*x^10 + ... G.f. = 2*q + 4*q^5 + 2*q^9 + 4*q^13 + 4*q^17 + 6*q^25 + 4*q^29 + 4*q^37 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x] / x^(1/4), {x, 0, n}]; (* Michael Somos, Feb 22 2015 *) s = 2*QPochhammer[q^2]^4/QPochhammer[q]^2+O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015 *) PROG (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); 2 * polcoeff( eta(x^2 A)^4 / eta(x + A)^2, n))}; (PARI) {a(n) = 2 * if( n<1, n==0, polcoeff( sum(k=0, (sqrtint(8*n + 1) - 1)\2, x^(k*(k + 1)/2), x*O(x^n))^2, n))}; CROSSREFS Cf. A004531, A008441. Sequence in context: A032059 A074075 A184186 * A246436 A143235 A069465 Adjacent sequences:  A004017 A004018 A004019 * A004021 A004022 A004023 KEYWORD nonn AUTHOR STATUS approved

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Last modified September 20 22:20 EDT 2018. Contains 315247 sequences. (Running on oeis4.)