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A005931
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Theta series of the coset of the E_7 lattice in its dual.
(Formerly M5313)
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1
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56, 576, 1512, 4032, 5544, 12096, 13664, 24192, 27216, 44352, 41832, 72576, 67536, 100800, 101304, 145728, 126504, 205632, 176456, 249984, 234360, 326592, 277200, 423360, 355320, 479808, 439992, 608832, 494928, 749952, 599760, 806400, 745416
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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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 (113)
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of 56* psi(q^2)^3* phi(q)^4 +128* q* psi(q^2)^7 in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos Jun 11 2007
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EXAMPLE
| 56*q^(3/2) + 576*q^(7/2) + 1512*q^(11/2) + 4032*q^(15/2) + 5544*q^(19/2) + ...
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PROG
| (PARI) {a(n)= local(A, B); if(n<0, 0, n++; A= sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n)); B= subst(A, x, -x); polcoeff( (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)), n))} /* Michael Somos Jun 11 2007*/
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CROSSREFS
| Sequence in context: A190070 A048540 A189497 * A027793 A055747 A027813
Adjacent sequences: A005928 A005929 A005930 * A005932 A005933 A005934
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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