A087401 Triangle of n*r-binomial(r+1,2).

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, 0, 10, 19, 27, 34, 40, 45, 49, 52, 54, 55, 55, 0, 11

Offset: 0

Author

Paul Boddington, Oct 21 2003

Keywords

• easy
• nonn
• tabl

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.

Examples

			  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
		

Links

• Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Crossrefs

Cf. A138666.

Programs

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

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 *)

Formula

T(0,0)=0 and for n>0: T(n,k)=T(n-1,k)+k for kReinhard Zumkeller, Oct 03 2012