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A005887 Theta series of f.c.c. lattice with respect to octahedral hole.
(Formerly M4070)
6
6, 8, 24, 0, 30, 24, 24, 0, 48, 24, 48, 0, 30, 32, 72, 0, 48, 48, 24, 0, 96, 24, 72, 0, 54, 48, 72, 0, 48, 72, 72, 0, 96, 24, 96, 0, 48, 56, 96, 0, 102, 72, 48, 0, 144, 48, 48, 0, 48, 72, 168, 0, 96, 72, 72, 0, 96, 48, 120, 0, 78, 48, 144, 0, 144, 120, 48, 0, 96, 72, 96, 0, 96, 56, 168 (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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..9999

G. Nebe and N. J. A. Sloane, Home page for this lattice

N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for sequences related to f.c.c. lattice

FORMULA

Expansion of q^(-1) * (phi^3(q) - phi^3(-q)) / 2 in powers of q^2 where phi() is a Ramanujan theta function. - Michael Somos, Aug 17 2009

A005875(2*n + 1) = a(n). - Michael Somos, Aug 17 2009

EXAMPLE

6 + 8*x + 24*x^2 + 30*x^4 + 24*x^5 + 24*x^6 + 48*x^8 + 24*x^9 + 48*x^

10 + ...

6*q + 8*q^3 + 24*q^5 + 30*q^9 + 24*q^11 + 24*q^13 + 48*q^17 + 24*q^19 + ...

MAPLE

maxd:=20001: read format: temp0:=trunc(evalf(sqrt(maxd)))+2: a:=0: for i from -temp0 to temp0 do a:=a+q^( (i+1/2)^2): od: th2:=series(a, q, maxd): a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od: th3:=series(a, q, maxd): th4:=series(subs(q=-q, th3), q, maxd):

t1:=series((th3^3-th4^3)/(2*q), q, maxd): t1:=series(subs(q=sqrt(q), t1), q, floor(maxd/2)): t2:=seriestolist(t1): for n from 1 to nops(t2) do lprint(n-1, t2[n]); od:

MATHEMATICA

s = (EllipticTheta[3, 0, q]^3 - EllipticTheta[3, 0, -q]^3)/(2q) + O[q]^200; CoefficientList[s, q^2] (* Jean-François Alcover, Sep 19 2016 *)

PROG

(PARI) {a(n) = if( n<0, 0, n = 2*n + 1; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1 + x*O(x^n))^3, n))} /* Michael Somos, Aug 17 2009 */

CROSSREFS

Cf. A005875.

Sequence in context: A034761 A085796 A280641 * A119875 A053189 A156231

Adjacent sequences:  A005884 A005885 A005886 * A005888 A005889 A005890

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 14 06:14 EST 2018. Contains 317162 sequences. (Running on oeis4.)