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Triangle T(n,k) = (k-1-n)*(k-2-n)*(k-2+2*n)/2 read by rows, 1<=k<=n.
2

%I #12 Dec 25 2018 13:10:05

%S 1,9,4,30,18,7,70,48,27,10,135,100,66,36,13,231,180,130,84,45,16,364,

%T 294,225,160,102,54,19,540,448,357,270,190,120,63,22,765,648,532,420,

%U 315,220,138,72,25,1045,900,756,616,483,360,250,156,81,28,1386,1210,1035,864,700,546,405,280,174,90,31

%N Triangle T(n,k) = (k-1-n)*(k-2-n)*(k-2+2*n)/2 read by rows, 1<=k<=n.

%C The triangle is defined as the matrix product A * B, A = [1; 1, 4; 1, 4, 7;...]; B = [1; 2, 1; 3, 2, 1;...]; both infinite lower triangular matrices with the rest of the terms zeros.

%e The first few rows of the triangle are:

%e 1;

%e 9, 4;

%e 30, 18, 7;

%e 70, 48, 27, 10;

%e 135, 100, 66, 36, 13;

%e 231, 180, 130, 84, 45, 16;

%e 364, 294, 225, 160, 102, 54, 19;

%e 540, 448, 357, 270, 190, 120, 63, 22;

%e 765, 648, 532, 420, 315, 220, 138, 72, 25;

%e 1045, 900, 756, 616, 483, 360, 250, 156, 81, 28;

%e 1386, 1210, 1035, 864, 700, 546, 405, 280, 174, 90, 31;

%e 1794, 1584, 1375, 1170, 972, 784, 609, 450, 310, 192, 99, 34, etc.

%p A104728 := proc(n)

%p (k-1-n)*(k-2-n)*(k-2+2*n)/2 ;

%p end proc:

%p seq(seq(A104728(n,k),k=1..n),n=1..14) ; # _R. J. Mathar_, Nov 07 2011

%t Table[(k-1-n)(k-2-n)(k-2+2n)/2,{n,20},{k,n}]//Flatten (* _Harvey P. Dale_, Dec 25 2018 *)

%Y Cf. A051798 (row sums), A007586, A002414 (column 1).

%K nonn,easy,tabl

%O 1,2

%A _Gary W. Adamson_, Mar 20 2005

%E Name contributed by _R. J. Mathar_, Nov 07 2011