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Search: id:A004013
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| A004013 |
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Theta series of body-centered cubic (b.c.c.) lattice.
(Formerly M4473)
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+0
4
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| 1, 0, 0, 8, 6, 0, 0, 0, 12, 0, 0, 24, 8, 0, 0, 0, 6, 0, 0, 24, 24, 0, 0, 0, 24, 0, 0, 32, 0, 0, 0, 0, 12, 0, 0, 48, 30, 0, 0, 0, 24, 0, 0, 24, 24, 0, 0, 0, 8, 0, 0, 48, 24, 0, 0, 0, 48, 0, 0, 72, 0, 0, 0, 0, 6, 0, 0, 24, 48, 0, 0, 0, 36, 0, 0, 56, 24, 0, 0, 0, 24, 0, 0, 72, 48, 0, 0, 0, 24, 0, 0
(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 and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan's theta function, Proc. Amer. Math. Soc., 128 (1999), 1333-1338; F_4(q).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 116.
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LINKS
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John Cannon, Table of n, a(n) for n = 0..10000
G. Nebe and N. J. A. Sloane, Home page for this lattice
Index entries for sequences related to b.c.c. lattice
Eric Weisstein's World of Mathematics, Theta Series
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FORMULA
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subs(q=q^2, ph)^3+(2*sqrt(q))^3*subs(q=q^4, ps)^3, where ps = A010054 = Sum_{k=0..infinity} q^(k*(k+1)/2), ph = A000122 = Sum_{k=-infinity, infinity} q^(k^2).
Expansion of phi(q^4)^3 +8*q^3*psi(q^8)^3 in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos Oct 25 2006
a(4n+1)=a(4n+2)=a(8n+7)=0.
Expansion of theta_3(q)^3 + theta_2(q)^3 in powers of q^(1/4).
G.f. is a period 1 Fourier series which satisfies f( -1 / (8 t)) = 2 (t/i)^(3/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A004015.
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EXAMPLE
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1 + 8*q^(3/2) + 6*q^2 + 12*q^4 + 24*q^(11/2) + 8*q^6 + 6*q^8 + 24*q^(19/2) + 24*q^10 + 24*q^12 + 32*q^(27/2) + ...
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MAPLE
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M:=100; M1:=M*(M+1)/2; ph:=series(add(q^(k^2), k=-M..M), q, M1): ps:=series(add(q^(k*(k+1)/2), k=0..M), q, M1): t1:=series(subs(q=q^2, ph)^3, q, M1): t2:=series((2*sqrt(q))^3*subs(q=q^4, ps)^3, q, M1): t3:=seriestolist(series(subs(q=q^2, t1+t2), q, M1)): for n from 0 to nops(t3)-1 do lprint(n, t3[n+1]); od:
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PROGRAM
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(PARI) {a(n)=if(n<0, 0, if(n%4==0, n/=4; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n))^3, n), if(n%8==3, n\=8; 8*polcoeff( sum(k=0, (sqrtint(8*n+1)-1)\2, x^((k^2+k)/2), x*O(x^n))^3, n))))} /* Michael Somos Oct 25 2006 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^8 + A)^5 / eta(x^4 + A)^2 / eta(x^16 + A)^2)^3 + (2 * x * eta(x^16 + A)^2 / eta(x^8 + A))^3, n))} /* Michael Somos May 17 2008 */
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CROSSREFS
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A005875(n)=a(4n). Cf. A004015.
Sequence in context: A010119 A010116 A031365 this_sequence A010118 A100121 A010526
Adjacent sequences: A004010 A004011 A004012 this_sequence A004014 A004015 A004016
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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