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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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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; refs; listen; history; internal format)
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OFFSET
| 0,4
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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
| 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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| John Cannon, Table of n, a(n) for n = 0..10000
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
G. Nebe and N. J. A. Sloane, Home page for this lattice
Eric Weisstein's World of Mathematics, Theta Series
Index entries for sequences related to b.c.c. lattice
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FORMULA
| 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
| 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
| 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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MATHEMATICA
| m = 13; m1 = m*((m + 1)/2); ph[q_] = Series[ Sum[ q^k^2, {k, -m, m}], {q, 0, m1}]; ps[q_] = Series[ Sum[ q^(k*((k + 1)/2)), {k, 0, m}], {q, 0, m1}]; t1[q_] = Normal[ Series[ ph[q^2]^3, {q, 0, m1}]]; t2[q_] = Normal[ Series[ (2*Sqrt[q])^3*ps[q^4]^3, {q, 0, m1}]]; CoefficientList[ Series[ t1[q^2] + t2[q^2], {q, 0, m1}], q] (* From Jean-François Alcover, Dec 20 2011, translated from Maple *)
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PROG
| (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
| A005875(n)=a(4n). Cf. A004015.
Sequence in context: A010119 A010116 A031365 * A010118 A100121 A010526
Adjacent sequences: A004010 A004011 A004012 * A004014 A004015 A004016
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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