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A004010 Theta series of 12-dimensional Coxeter-Todd lattice K_12.
(Formerly M5478)
7
1, 0, 756, 4032, 20412, 60480, 139860, 326592, 652428, 1020096, 2000376, 3132864, 4445532, 7185024, 10747296, 13148352, 21003948, 27506304, 33724404, 48009024, 64049832, 70709184, 102958128, 124782336, 142254252, 189423360, 237588120, 248250240, 344391264 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 129.

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

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..500 from N. J. A. Sloane)

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

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

Eric Weisstein's World of Mathematics, Coxeter-Todd Lattice

Eric Weisstein's World of Mathematics, Theta Series

FORMULA

G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 27 (t/i)^6 f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 20 2015

G.f.: (3*a(x)^6 - 4*a(x)^3*b(x)^3 + 4*b(x)^6) / 3 where a(), b() are cubic AGM theta functions. - Michael Somos, Dec 25 2015

EXAMPLE

G.f. = 1 + 756*x^2 + 4032*x^3 + 20412*x^4 + 604890*x^5 + 139860*x^6 + ...

G.f. = 1 + 756*q^4 + 4032*q^6 + 20412*q^8 + 60480*q^10 + 139860*q^12 + 326592*q^14 + 652428*q^16 + 1020096*q^18 + 2000376*q^20 + ...

MAPLE

# Jacobi theta constants th2, th3: maxd := 201: 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);

# get phi0 and phi1: phi0 := series( subs(q=q^2, th2)*subs(q=q^6, th2)+subs(q=q^2, th3)*subs(q=q^6, th3), q, maxd); phi1 := series( subs(q=q^2, th2)*subs(q=q^6, th3)+subs(q=q^2, th3)*subs(q=q^6, th2), q, maxd);

K_12 := series( subs(q=q^2, phi0)^6+45*subs(q=q^2, phi0)^2*subs(q=q^2, phi1)^4+18*subs(q=q^2, phi1)^6, q, maxd);

MATHEMATICA

maxd = 51; temp0 = Floor[ Sqrt[maxd] ]+2; a = 0; Do[ a=a+q^(i+1/2)^2, {i, -temp0, temp0}]; th2[q_] = Normal[ Series[a, {q, 0, maxd}]]; a = 0; Do[ a=a+q^i^2, {i, -temp0, temp0}]; th3[q_] = Normal[ Series[a, {q, 0, maxd}]]; phi0[q_] = Normal[ Series[ th2[q^2]*th2[q^6] + th3[q^2]*th3[q^6], {q, 0, maxd}]]; phi1[q_] = Normal[ Series[ th2[q^2]*th3[q^6] + th3[q^2]*th2[q^6], {q, 0, maxd}]]; K12 = Series[ phi0[q^2]^6 + 45*phi0[q^2]^2*phi1[q^2]^4 + 18*phi1[q^2]^6, {q, 0, maxd}]; CoefficientList[ K12, q^2 ] (* Jean-François Alcover, Nov 28 2011, translated from Maple *)

a[ n_] := With[ {U1 = QPochhammer[ q]^3, U3 = QPochhammer[ q^3]^3, U9 = QPochhammer[ q^9]^3}, With[ {z = (1 + 9 q U9/U1)^3}, SeriesCoefficient[ (U1^3/U3)^2 (3 z^2 - 4 z + 4) / 3, {q, 0, n}]]]; (* Michael Somos, Dec 25 2015 *)

PROG

(MAGMA) A := Basis( ModularForms( Gamma1(3), 6), 29); A[1] + 756*A[3]; /* Michael Somos, Dec 20 2015 */

(PARI) {a(n) = my(A, U1, U3, U9, z); if( n<0, 0, A = x * O(x^n); U1 = eta(x + A)^3; U3 = eta(x^3 + A)^3; U9 = eta(x^9 + A)^3; z = (1 + 9 * x * U9/U1)^3; polcoeff( (U1^3/U3)^2 * (3*z^2 - 4*z +4) / 3, n))}; /* Michael Somos, Dec 25 2015 */

CROSSREFS

Cf. A004046, A008657, A107658.

Sequence in context: A252007 A202198 A158657 * A320686 A339797 A288779

Adjacent sequences:  A004007 A004008 A004009 * A004011 A004012 A004013

KEYWORD

easy,nonn,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 30 20:17 EST 2021. Contains 349425 sequences. (Running on oeis4.)