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Search: id:A004046
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| A004046 |
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Theta series of extremal 3-modular even 24-dimensional lattice with minimal norm 6 and det = 3^12. |
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5
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| 1, 0, 0, 26208, 530712, 6368544, 47331648, 256864608, 1116087336, 4092877152, 12996075456, 37058557536, 96952754808, 232778774592, 526258264896, 1128148021728, 2286143305992, 4451523096384
(list; graph; listen)
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OFFSET
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0,4
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REFERENCES
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N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.
H.-G. Quebbemann, Modular lattices in Euclidean spaces, J. Number Theory, 54 (1995), 190-202.
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LINKS
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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 lattice
N. J. A. Sloane, Seven Staggering Sequences.
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FORMULA
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Theta series = a^12 - 9/2*a^8*b^4 + 414*a^6*b^6 + 1458*a^4*b^8 + 1998*a^2*b^10 + 459/2*b^12 (see PARI code for details)
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PROGRAM
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(PARI) th3 = sum(n=1, noo\2, 2*x^(4*n^2), 1+A);
(PARI) th4 = sum(n=1, noo\2, (-1)^n*2*x^(4*n^2), 1+A);
(PARI) th2 = sum(n=0, noo\2, 2*x^(4*n^2+4*n+1), A);
(PARI) chk("th3^4 == th4^4+th2^4");
(PARI) /* A004016(x^4) */
(PARI) phi0 = th2*subst(th2, x, x^3)+ th3*subst(th3, x, x^3);
(PARI) /* 2*x*A033762(x^2) */
(PARI) phi1 = th2*subst(th3, x, x^3)+ th3*subst(th2, x, x^3);
(PARI) /* A004010(x^2) */
(PARI) K_12 = phi0^6+45*phi0^2*phi1^4+18*phi1^6;
(PARI) a=phi0; b=phi1;
(PARI) A004046=a^12-9/2*a^8*b^4+414*a^6*b^6+1458*a^4*b^8+1998*a^2*b^10+459/2*b^12;
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CROSSREFS
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Cf. A107657.
Sequence in context: A003927 A115494 A034622 this_sequence A107119 A015303 A046710
Adjacent sequences: A004043 A004044 A004045 this_sequence A004047 A004048 A004049
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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PARI code from Michael Somos, Jun 07 2005
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