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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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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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njas
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