Search: id:a045828
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A045828
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One fourth of theta series of cubic lattice with respect to face.
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+0
20
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1, 2, 2, 4, 3, 2, 6, 4, 4, 6, 4, 4, 7, 8, 2, 8, 8, 4, 10, 4, 4, 10, 10, 8, 9, 4, 6, 12, 8, 6, 10, 12, 4, 14, 8, 4, 16, 10, 8, 8, 9, 10, 12, 12, 8, 12, 12, 4, 20, 10, 6, 20, 8, 6, 10, 12, 8, 20, 18, 8, 11, 12, 12, 16, 8, 6, 20, 16, 12, 14, 8, 12, 20, 14, 6, 12, 20, 8, 26, 12, 8, 22, 8, 12, 15
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OFFSET
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0,2
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COMMENTS
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Number of solutions to n = t1 + t2 + 2*t3 where t1, t2, t3 are triangular numbers. - Michael Somos, Jan 02 2006
The cubic lattice is the set of triples [a, b, c] where the entries are all integers. A face is centered at a triple where one entry is an integer and the other two are one half an odd integer. - Michael Somos, Jun 29 2012
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.
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LINKS
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FORMULA
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Expansion of q^(-1/2) * (eta(q^2)^3 * eta(q^4)^2) / eta(q)^2 in powers of q. - Michael Somos, Jan 02 2006
Expansion of phi(x) * psi(x^2)^2 = psi(x)^2 * psi(x^2) = psi(x)^4 / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jun 29 2012
Euler transform of period 4 sequence [2, -1, 2, -3, ...]. - Michael Somos, Mar 05 2003
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = (1/2)^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A212885. - Michael Somos, Sep 08 2018
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EXAMPLE
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G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 3*x^4 + 2*x^5 + 6*x^6 + 4*x^7 + 4*x^8 + 6*x^9 + ...
G.f. = q + 2*q^3 + 2*q^5 + 4*q^7 + 3*q^9 + 2*q^11 + 6*q^13 + 4*q^15 + 4*q^17 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 1/4 EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x]^2, {x, 0, n + 1/2}]; (* Michael Somos, Jun 29 2012 *)
a[ n_] := SeriesCoefficient[ 1/8 EllipticTheta[ 2, 0, x^2] EllipticTheta[ 2, 0, x]^2, {x, 0, 2 n + 1}]; (* Michael Somos, Jun 29 2012 *)
QP = QPochhammer; s = (QP[q^2]^3*QP[q^4]^2)/QP[q]^2 + O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A)^2 / eta(x + A)^2, n))}; /* Michael Somos, Oct 25 2006 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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