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A004010
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Theta series of 12-dimensional Coxeter-Todd lattice K_12.
(Formerly M5478)
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3
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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).
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 0..500
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
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EXAMPLE
| 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 + ...
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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);
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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 ] (* From Jean-François Alcover, Nov 28 2011, translated from Maple *)
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CROSSREFS
| Cf. A004046, A107658.
Sequence in context: A108374 A202198 A158657 * A018898 A004048 A037191
Adjacent sequences: A004007 A004008 A004009 * A004011 A004012 A004013
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KEYWORD
| easy,nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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