OFFSET
0,8
COMMENTS
There is a curious connection with the character tables of cyclic groups of prime power order. Let G be a cyclic group of order p^n where p is prime and n is nonnegative. Construct an (n+1)x(n+1) matrix A whose rows and columns are indexed by the set 0,1,...,n as follows. The ij entry is obtained by taking any element of order p^(n-j) in G and summing its character values over all characters of order p^i in the dual group of G. Remarkably, all coefficients of the characteristic polynomial of A are powers of p (with alternating signs) and these powers can be read off from the appropriate row of our triangle. For example if n=2 then the characteristic polynomial is X^3 - p^2*X^2 + p^3*X - p^3.
LINKS
Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
FORMULA
T(0,0)=0 and for n>0: T(n,k)=T(n-1,k)+k for k<n and T(n,n)=T(n,n-1). - Reinhard Zumkeller, Oct 03 2012
EXAMPLE
0
0 0
0 1 1
0 2 3 3
0 3 5 6 6
0 4 7 9 10 10
0 5 9 12 14 15 15
0 6 11 15 18 20 21 21
0 7 13 18 22 25 27 28 28
0 8 15 21 26 30 33 35 36 36
0 9 17 24 30 35 39 42 44 45 45
MAPLE
A087401 := proc(n, k)
n*k-binomial(k+1, 2) ;
end proc:
seq(seq( A087401(n, k), k=0..n), n=0..12) ; # R. J. Mathar, Jan 21 2015
MATHEMATICA
Table[n*r-Binomial[r+1, 2], {n, 0, 20}, {r, 0, n}]//Flatten (* Harvey P. Dale, Jul 10 2020 *)
PROG
(Haskell)
a087401 n k = a087401_tabl !! n !! k
a087401_row n = a087401_tabl !! n
a087401_tabl = iterate f [0] where
f row = row' ++ [last row'] where row' = zipWith (+) row [0..]
-- Reinhard Zumkeller, Oct 03 2012
CROSSREFS
KEYWORD
AUTHOR
Paul Boddington, Oct 21 2003
STATUS
approved