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A005876 Theta series of cubic lattice with respect to edge.
(Formerly M1824)
5
2, 8, 10, 8, 16, 16, 10, 24, 16, 8, 32, 24, 18, 24, 16, 24, 32, 32, 16, 32, 34, 16, 48, 16, 16, 56, 32, 24, 32, 40, 26, 48, 48, 16, 32, 32, 32, 56, 48, 24, 64, 32, 26, 56, 16, 40, 64, 64, 16, 40, 48, 32, 80, 32, 32, 64, 50, 40, 48, 48, 48, 56, 48, 16, 64, 72, 32, 88, 32, 24 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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).

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.

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

a(n) = 2*A045834(n).

Expansion of 2*phi(q)*psi(q)^2 in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos, Feb 21 2006

Expansion of theta_2(q^2)^2(theta_3(q)+theta_4(q))/(4q) in powers of q^4. - Michael Somos, Feb 21 2006

Expansion of 2q^(-1/4)eta(q^2)^9/(eta(q)^4*eta(q^4)^2) in powers of q. - Michael Somos, Feb 21 2006

G.f.: 2*Product_{k>0} (1+x^k)^4*(1-x^(2k))^3/(1+x^(2k))^2. - Michael Somos, Feb 21 2006

MATHEMATICA

s = EllipticTheta[3, 0, q]^2*EllipticTheta[2, 0, q]/q^(1/4) + O[q]^70; CoefficientList[s, q] (* Jean-François Alcover, Nov 04 2015, from 1st formula *)

s = (2*(QPochhammer[q^2]^9/(QPochhammer[q]^4*QPochhammer[q^4]^2))) + O[q]^70; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015, from 3rd formula *)

PROG

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); 2*polcoeff( eta(x^2+A)^9/ eta(x+A)^4/eta(x^4+A)^2, n))} /* Michael Somos, Feb 21 2006 */

CROSSREFS

Cf. A045834.

Sequence in context: A036898 A053372 A088155 * A139370 A101532 A032708

Adjacent sequences:  A005873 A005874 A005875 * A005877 A005878 A005879

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 19 15:53 EST 2019. Contains 319307 sequences. (Running on oeis4.)