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A014453
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Theta series of quadratic form with Gram matrix [ 2, 0, 0; 0, 2, 1; 0, 1, 2 ].
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2
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1, 8, 12, 6, 20, 24, 0, 24, 36, 8, 24, 24, 18, 48, 24, 0, 44, 48, 12, 24, 48, 24, 48, 48, 0, 56, 24, 6, 72, 72, 24, 24, 84, 0, 24, 48, 20, 96, 48, 24, 72, 48, 0, 72, 72, 24, 48, 48, 42, 56, 60, 0, 96, 120, 0, 48, 72, 48, 72, 24, 0, 96, 72, 24, 92, 96, 24, 72, 120, 0, 48, 48, 36, 96, 72
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OFFSET
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0,2
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COMMENTS
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This is the hexagonal P lattice (the even holotype) of dimension 3.
a(n) is the number of solutions to x^2 + y^2 + z^2 + x*y = n in integers. - Michael Somos, Jul 03 2018
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LINKS
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FORMULA
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Expansion of a(x) * phi(x) where phi() is a Ramanujan theta function and a() is a cubic AGM theta function. - Michael Somos, May 30 2012
Expansion of (eta(q)^3 + 9 * eta(q^9)^3) * eta(q^2)^5 / (eta(q)^2 * eta(q^3) * eta(q^4)^2) in powers of q.
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EXAMPLE
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G.f. = 1 + 8*x + 12*x^2 + 6*x^3 + 20*x^4 + 24*x^5 + 24*x^7 + 36*x^8 + 8*x^9 + ...
G.f. = 1 + 8*q^2 + 12*q^4 + 6*q^6 + 20*q^8 + 24*q^10 + 24*q^14 + 36*q^16 + 8*q^18 + ...
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MATHEMATICA
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(* A004016 *) a2[0] = 1; a2[n_] := 6*DivisorSum[n, KroneckerSymbol[#, 3]&]; (* A000122 *) a3[n_] := SeriesCoefficient[EllipticTheta[3, 0, q], {q, 0, n}]; a[n_] := Sum[a2[k]*a3[n-k], {k, 0, n}]; Table[a[n], {n, 0, 75}] (* Jean-François Alcover, Nov 04 2015, from the convolution given by Michael Somos *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] (QPochhammer[ x]^3 + 9 x QPochhammer[ x^9]^3) / QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Jul 03 2018 *)
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PROG
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(PARI) {a(n) = if( n<1, n==0, 2 * qfrep( [ 2, 0, 0; 0, 2, 1; 0, 1, 2 ], n, 1)[n])}; /* Michael Somos, May 30 2012 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3) / eta(x^3 + A) * eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^4 + A)^2), n))}; /* Michael Somos, May 30 2012 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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