A129548 -id:A129548 - cp's OEIS frontend

A259181 a(n) = n(n+1)(n+2)(n+3)(2n^2+6n+7)/360.

0, 1, 9, 43, 147, 406, 966, 2058, 4026, 7359, 12727, 21021, 33397, 51324, 76636, 111588, 158916, 221901, 304437, 411103, 547239, 719026, 933570, 1198990, 1524510, 1920555, 2398851, 2972529, 3656233, 4466232, 5420536, 6539016, 7843528, 9358041, 11108769

Offset: 0

Luce ETIENNE, Nov 08 2015

After 0, second bisection of A129548.

This sequence is also the total number of squares of all sizes in i X i subsquares in an n X n grid, whereas A000330 simply gives the number of all sizes of squares in an n X n grid. See illustrations.

			a(0) = 0; a(1) = 1*1; a(2) = 4*1+1*5 = 9; a(3) = 9*1+4*5+1*14 = 43.

Colin Barker, Table of n, a(n) for n = 0..1000
Luce ETIENNE, illustration of initial terms
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 14.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000217, A000290, A000330, A000539, A129548.

Cf. A060060: (1/6)*Sum_{i=0..n} (i+1)*(i+2)*(2*i+3)*i^2.

vector(100, n, n--; n*(n+1)*(n+2)*(n+3)*(2*n^2+6*n+7)/360) \\ Altug Alkan, Nov 08 2015

concat(0, Vec(-x*(x+1)^2 / (x-1)^7 + O(x^100))) \\ Colin Barker, Nov 08 2015

a(n) = (1/6)*Sum_{i=0..n} (i+1)*(i+2)*(2*i+3)*(n-i)^2.
a(n) = Sum_{i=0..n} A000290(n-i)*A000330(i+1).
G.f.: x*(1 + x)^2 / (1 - x)^7. - Colin Barker, Nov 08 2015
a(n) = (A000539(n+1) - A000217(n+1))/30. - Yasser Arath Chavez Reyes, Feb 24 2024

A129549 Dimension of space of measures of entanglement that are homogeneous of degree 2n, for the case of four qubits.

Original entry on oeis.org

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

Mike Zabrocki, Apr 20 2007

nonn

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.

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.

Cf. A000217, A129548.

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

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.

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

A129550 Number of real polynomial invariants for the action of 4 copies of U(2) on the fourth tensor power of C^2.

Original entry on oeis.org

1, 1, 8, 20, 98, 293, 1128, 3409, 10846, 30480, 84652, 217677, 544312, 1289225, 2961626, 6528284, 13980717, 28963980, 58464510, 114806429, 220298632, 412950779, 758418342, 1365044296, 2412766496, 4189995629, 7159916414

Offset: 0

Mike Zabrocki, Apr 20 2007

nonn

M. W. Hero and J. F. Willenbring, Stable Hilbert series as related to the measurement of quantum entanglement, Discrete Math., 309 (2010), 6508-6514.
Nolan R. Wallach, The Hilbert series for measures of entanglement in 4 qubits, Acta Appl. Math. 86(2005),203-220.

Cf. A129548, A129549.

G.f.: (R(q) + q^76*R(1/q))/((1 - q^2)*(1 - q^4)^7*(1 - q^6)^6*(1 - q^8)^4*(1 - q^10)) where R(q) = 1 + 6*q^6 + 46*q^8 + 110*q^10 + 344*q^12 + 844*q^14 + 2154*q^16 + 4606*q^18 + 9397*q^20 + 16848*q^22 + 28747*q^24 + 44580*q^26 + 65366*q^28 + 88036*q^30 + 111909*q^32 + 131368*q^34 + 145676*q^36 + 149860/2*q^38.

cp's OEIS Frontend

A259181 a(n) = n(n+1)(n+2)(n+3)(2n^2+6n+7)/360.

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A129549 Dimension of space of measures of entanglement that are homogeneous of degree 2n, for the case of four qubits.

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A129550 Number of real polynomial invariants for the action of 4 copies of U(2) on the fourth tensor power of C^2.

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Author

Keywords

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Formula

cp's OEIS Frontend

A259181 a(n) = n*(n+1)*(n+2)*(n+3)*(2*n^2+6*n+7)/360.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A129549 Dimension of space of measures of entanglement that are homogeneous of degree 2n, for the case of four qubits.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Formula

Extensions

A129550 Number of real polynomial invariants for the action of 4 copies of U(2) on the fourth tensor power of C^2.

Views

Author

Keywords

References

Crossrefs

Formula

A259181 a(n) = n(n+1)(n+2)(n+3)(2n^2+6n+7)/360.