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A004008
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Expansion of theta series of E_7 lattice in powers of q^2.
(Formerly M5388)
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1
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1, 126, 756, 2072, 4158, 7560, 11592, 16704, 24948, 31878, 39816, 55944, 66584, 76104, 99792, 116928, 133182, 160272, 177660, 205128, 249480, 265104, 281736, 350784, 382536, 390726, 470232, 505568, 532800, 615384, 640080, 701568, 799092
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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REFERENCES
| J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 125. Equation (112)
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| G. Nebe and N. J. A. Sloane, Home page for this lattice
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of phi(q)^3*(phi(q)^4 +7*16*q*psi(q^2)^4) in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos Oct 24 2006
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EXAMPLE
| 1 + 126*q^2 + 756*q^4 + 2072*q^6 + 4158*q^8 + 7560*q^10 + 11592*q^12 + ...
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PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n)); polcoeff( 8*A^7 -7*A^3*subst(A, x, -x)^4, n))} /* Michael Somos Oct 24 2006 */
(PARI) {a(n)= if(n<1, n==0, 2*qfrep([2, -1, 0, 0, 0, 0, 0; -1, 2, -1, 0, 0, 0, 0; 0, -1, 2, -1, 0, 0, 0; 0, 0, -1, 2, -1, 0, -1; 0, 0, 0, -1, 2, -1, 0; 0, 0, 0, 0, -1, 2, 0; 0, 0, 0, -1, 0, 0, 2], n, 1)[n])} /* Michael Somos Jun 11 2007 */
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CROSSREFS
| Sequence in context: A186817 A107658 A181254 * A126170 A151989 A104678
Adjacent sequences: A004005 A004006 A004007 * A004009 A004010 A004011
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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