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

Offset: 0

Author

Mike Zabrocki, Apr 20 2007

Keywords

• nonn

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.

Crossrefs

Cf. A000217, A129548.

Programs

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

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.

Extensions

Revised definition from N. J. A. Sloane, Jun 17 2011