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A087401 Triangle of n*r-binomial(r+1,2). 5
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 (list; table; graph; refs; listen; history; text; internal format)
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

Cf. A138666.

Sequence in context: A340261 A190146 A026932 * A332915 A192498 A308178

Adjacent sequences: A087398 A087399 A087400 * A087402 A087403 A087404

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Boddington, Oct 21 2003

STATUS

approved

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Last modified December 5 01:43 EST 2022. Contains 358572 sequences. (Running on oeis4.)