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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

M. Somos, Introduction to Ramanujan theta functions

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-Fran├žois 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

N. J. A. Sloane

STATUS

approved

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Last modified June 24 01:22 EDT 2018. Contains 311819 sequences. (Running on oeis4.)