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A129549

Dimension of space of measures of entanglement that are homogeneous of degree 2n, for the case of four qubits.

1, 3, 20, 78, 352, 1365, 5232, 18271, 60598, 187296, 548020, 1515265, 3991204, 10035401, 24210308, 56188768, 125904351, 273044682, 574635828, 1176027747, 2345376048, 4565886531, 8691118644, 16198834634, 29602895824, 53105875363

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OFFSET

0,2

REFERENCES

David Meyer and Nolan Wallach, Invariants for multiple qubits: the case of 3 qubits, Mathematics of quantum computing, Computational Mathematics Series, 77-98, Chapman&Hall/CRC, 2002.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Nolan Wallach, The Hilbert series of measures of entanglement for 4 q-bits, Acta Appl. Math. 86(2005), 203-220.

FORMULA

a(n) = [q^(2n)] (P(q) + q^54*P(1/q))/((1 - q^2)^3*(1 - q^4)^11*(1 - q^6)^6) where P(q) = 1 + 3*q^4 + 20*q^6 + 76*q^8 + 219*q^10 + 654*q^12 + 1539*q^14 + 3119*q^16 + 5660*q^18 + 9157*q^20 + 12876*q^22 + 16177*q^24 + 18275*q^26.

MAPLE

t1:=1 + 3*q^4 + 20*q^6 + 76*q^8 + 219*q^10 + 654*q^12 +

1539*q^14 + 3119*q^16 + 5660*q^18 + 9157*q^20 +

12876*q^22 + 16177*q^24 + 18275*q^26 +

18275*q^28 + 16177*q^30 + 12876*q^32 +

9157*q^34 + 5660*q^36 + 3119*q^38 + 1539*q^40 +

654*q^42 + 219*q^44 + 76*q^46 + 20*q^48 + 3*q^50 + q^54;

t2:=(1-q^2)^3*(1-q^4)^11*(1-q^6)^6;

t3:=t1/t2;

t4:=subs(q=sqrt(x), t3);

t5:=series(t4, x, 30); # N. J. A. Sloane, Jun 17 2011

CROSSREFS