1,2

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.rs), 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: A064503 A050482 A153765 this_sequence A048657 A000639 A052528

Adjacent sequences: A003603 A003604 A003605 this_sequence A003607 A003608 A003609

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein

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