A138544 -id:A138544 - cp's OEIS frontend

A138545 Central moment sequence of tr(A^4) in USp(6).

1, 0, 3, 1, 27, 26, 385, 708, 7231, 20296, 164277, 608565, 4286161, 19021302, 123867107, 617758729, 3862576095, 20774382552, 127548675709, 720773229015, 4401180707397, 25709943020830, 157204921750191, 939751281408962

Offset: 0

Andrew V. Sutherland, Mar 24 2008

nonn

If A is a random matrix in the compact group USp(6) (6x6 complex matrices which are unitary and symplectic), then a(n)=E[(tr(A^4+1))^n] is the n-th central moment of the trace of A^4, since E[tr(A^4)] = -1 (see A138544).

			a(5) = 26 because E[(tr(A^4)+1)^5] = 26 for a random matrix A in USp(6).

Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.

Cf. A138544.

mgf is A(z)=e^zF(z) where F(z) is the mgf of A138544.

A309838 a(n) = number of dimensions of semisimple matrix subalgebras.

Original entry on oeis.org

2, 4, 7, 11, 16, 22, 29, 39, 50, 60, 73, 88, 103, 120, 139, 160, 181, 203, 229, 256, 284, 313, 343, 377, 412, 448, 487, 528, 569, 610, 653, 699, 748, 797, 849, 904, 959, 1014, 1070, 1129, 1191, 1255, 1321, 1388, 1456, 1526, 1598, 1672, 1746, 1821, 1899, 1981, 2064

Offset: 1

Phillip Heikoop, Aug 19 2019

nonn

a(n) = |A_n| in the Heikoop paper.

A semisimple matrix subalgebra of M_n(k) for an algebraically closed field k is a direct sum of M_n_i(k) such that Sum (n_i) <= n. See Heikoop paper, Section 3.2, for more.

Phillip Heikoop, Table of n, a(n) for n = 1..336
Phillip Tomas Heikoop, Dimensions of Matrix Subalgebras, Bachelor's Thesis, Worcester Polytechnic Institute (2019).
Phillip Heikoop, C++11 code to generate the sequence

Cf. A000124, A002620, A069999, A138544, A309839.

a(n) <= n^2 - Sqrt(2) * Sqrt(2n+ 3) * n.

A309839 a(n) = GAP_n: first integer m that is not the dimension of a semisimple subalgebra of M_n(k).

Original entry on oeis.org

3, 6, 7, 12, 15, 22, 23, 42, 43, 48, 63, 76, 79, 96, 115, 140, 143, 166, 167, 192, 247, 248, 279, 312, 347, 384, 423, 472, 483, 526, 527, 572, 619, 624, 719, 724, 827, 832, 889, 948, 1009, 1072, 1087, 1152, 1219, 1288, 1359, 1432, 1507, 1520, 1597, 1676, 1679

Offset: 2

Phillip Heikoop, Aug 19 2019

nonn

Define the sequence a(n) = GAP_n to be the smallest integer that is not the dimension of a semisimple subalgebra of M_n(k). This is one more than the upper endpoint of the continuous region of M_n(k). Because when n = 1 there are no gaps, this sequence begins at n = 2. See Heikoop paper, page 31.

Michael De Vlieger, Table of n, a(n) for n = 2..101
Phillip Tomas Heikoop, Dimensions of Matrix Subalgebras, Bachelor's Thesis, Worcester Polytechnic Institute (2019).
Phillip Heikoop, C++11 code to generate the sequence

Cf. A000124, A002620, A069999, A138544, A309838.

a(n) > n^2 - 4 * sqrt(n + 2).

cp's OEIS Frontend

A138545 Central moment sequence of tr(A^4) in USp(6).

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A309838 a(n) = number of dimensions of semisimple matrix subalgebras.

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A309839 a(n) = GAP_n: first integer m that is not the dimension of a semisimple subalgebra of M_n(k).

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