1,2

J. A. Green, The characters of the finite general linear groups, Trans. Amer. Math. Soc., 80 (1955), 402-447.

R. Steinberg, A geometric approach to the representations of the full linear group over a Galois field, Trans. Amer. Math. Soc., 71 (1951), 274-282.

T. D. Noe, Table of n, a(n) for n=1..500

N. J. A. Sloane, Transforms

G.f.: Product_{k >= 1} f(x^k)^p_k, where f(x)=Product_{k >= 0} 1/(1-x^k) = Sum_{k >= 0} p_k*x^k and p_k is the number of partitions of k (A000041).

Recurrence relation: a(n+1) = 1/(n+1) * Sum_{0=<k<=n} a(k)*g(n-k+1) where g(n) = Sum_{ij | n} p(i)*i*j, with the sum over all ordered pairs (i, j) such that their products divide n, and p(i) is the number of partitions of i. Also a(0)=1.

Euler transform of A047968(n). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jun 23 2004

Recurrence relation: a(0)=1, a(n+1) = 1/(n+1) * Sum_{0=<k<=n} a(k)*g(n-k+1) where g(n) = Sum_{d | n} d * A000041(d) * A000203(n/d). - Brett Witty (witty(AT)maths.anu.edu.au), Jul 12 2006

a(2) = 4 as there are four types of conjugacy classes of 2 x 2 matrices over GF(q):

* the scalar matrices (diagonal matrix with both entries the same)

* the direct sum of two scalars (diagonal matrix with both entries different)

* the non-diagonalizable Jordan block (upper triangular matrix with the same entry along the diagonal and a 1 in the superdiagonal)

* companion matrices of irreducible quadratics over GF(q)

This example can be found in Green's paper (in the references).

(GAP) a := function(n) local k, sum; sum := 0; for k in [0..n-1] do sum := sum + a(k)*g(n-k); od; return sum/n; end;

g := function(n) local i, j, sum; for i in DivisorsInt(n) do for j in DivisorsInt(n/i) do sum := sum + NrPartitions(i)*i*j; od; od; return sum; end; ;

# This code is significantly faster if you store previously computed values of a(n) and g(n).

(GAP) a := function(n) if( n = 0) then return 1; else return Sum([0..n], i -> t(i) * Sum(DivisorsInt(n-i), d -> d * NrPartitions(d) * Sigma(n/d)) )/n; fi; end; ; - Brett Witty (witty(AT)maths.anu.edu.au), Jul 12 2006

Cf. A001970.

Cf. A006951, A006952, A049314, A049315, A049316.

Sequence in context: A129794 A064503 A050482 this_sequence A048657 A000639 A052528

Adjacent sequences: A003603 A003604 A003605 this_sequence A003607 A003608 A003609

nonn,nice,easy

njas, Mira Bernstein

More terms, recurrence and GAP program from Brett Witty (witty(AT)maths.anu.edu.au), Jul 17 2003