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

%I A087401 #17 Jul 10 2020 14:08:38
%S A087401 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,
%T A087401 15,18,20,21,21,0,7,13,18,22,25,27,28,28,0,8,15,21,26,30,33,35,36,36,
%U A087401 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
%N A087401 Triangle of n*r-binomial(r+1,2).
%C A087401 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.
%H A087401 Reinhard Zumkeller, <a href="/A087401/b087401.txt">Rows n = 0..120 of triangle, flattened</a>
%F A087401 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
%e A087401   0
%e A087401   0   0
%e A087401   0   1   1
%e A087401   0   2   3   3
%e A087401   0   3   5   6   6
%e A087401   0   4   7   9  10  10
%e A087401   0   5   9  12  14  15  15
%e A087401   0   6  11  15  18  20  21  21
%e A087401   0   7  13  18  22  25  27  28  28
%e A087401   0   8  15  21  26  30  33  35  36  36
%e A087401   0   9  17  24  30  35  39  42  44  45  45
%p A087401 A087401 := proc(n,k)
%p A087401     n*k-binomial(k+1,2) ;
%p A087401 end proc:
%p A087401 seq(seq( A087401(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Jan 21 2015
%t A087401 Table[n*r-Binomial[r+1,2],{n,0,20},{r,0,n}]//Flatten (* _Harvey P. Dale_, Jul 10 2020 *)
%o A087401 (Haskell)
%o A087401 a087401 n k = a087401_tabl !! n !! k
%o A087401 a087401_row n = a087401_tabl !! n
%o A087401 a087401_tabl = iterate f [0] where
%o A087401    f row = row' ++ [last row'] where row' = zipWith (+) row [0..]
%o A087401 -- _Reinhard Zumkeller_, Oct 03 2012
%Y A087401 Cf. A138666.
%K A087401 easy,nonn,tabl
%O A087401 0,8
%A A087401 _Paul Boddington_, Oct 21 2003