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Search: id:A003781
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| A003781 |
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Expansion of theta series of {E_7}* lattice in powers of q^(1/2). |
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3
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| 1, 0, 0, 56, 126, 0, 0, 576, 756, 0, 0, 1512, 2072, 0, 0, 4032, 4158, 0, 0, 5544, 7560, 0, 0, 12096, 11592, 0, 0, 13664, 16704, 0, 0, 24192, 24948, 0, 0, 27216, 31878, 0, 0, 44352, 39816, 0, 0, 41832, 55944, 0, 0, 72576, 66584, 0, 0, 67536, 76104, 0, 0, 100800
(list; graph; listen)
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OFFSET
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0,4
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 125.
M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhauser, 1985, p. 141.
N. Elkies and B. H. Gross, Embeddings into the integral octonions, Olga Taussky-Todd: in memoriam, Pacific J. Math. 1997, Special Issue, 147-158.
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FORMULA
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Theta series is given on page 125 of Conway and Sloane.
Can be determined from A023919 (A*_7): [1] A003781(4n)=A023919(16n) [2] A003781(4n+3)=A023919(16n+12). Let A_7+[1] be the generator of A*_7/A_7, then these correspond to [1]A004008=theta(E_7)=theta(A_7)+theta(A_7+[4]), [2]A005931=theta(E_7+[1])=theta(A_7+[2])+theta(A_7+[6]) - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 03 2000
a(4n+1) = a(4n+2) = 0. - Michael Somos Jun 11 2007
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EXAMPLE
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1 + 56*q^(3/2) + 126*q^2 + 576*q^(7/2) + 756*q^4 + 1512*q^(11/2) + ...
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PROGRAM
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(PARI) {a(n)= local(A, B, m); n++; m=n%4; n\=4; if(n<0 | m>1, 0, A= sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n)); B= subst(A, x, -x); polcoeff( if(m==0, (A^4 -B^4)* (8*A^4 -B^4)/ 2/ sum(k=0, sqrtint( 4*n+1)\2, x^(k^2+k), x*O(x^n)), 8*A^7 -7*A^3*subst(A, x, -x)^4 ), n))} /* Michael Somos Jun 11 2007 */
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CROSSREFS
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Cf. A030443, A038723.
a(4n) = A004008(n), a(4n+3) = A005931(n). - Michael Somos Jun 11 2007.
Sequence in context: A044624 A157330 A038849 this_sequence A030443 A135803 A048452
Adjacent sequences: A003778 A003779 A003780 this_sequence A003782 A003783 A003784
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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