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Triangle T(n,k) = k*(k-n-1)*(k-n-2)/2 read by rows, 1<=k<=n.
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%I #22 May 22 2023 08:55:02

%S 1,3,2,6,6,3,10,12,9,4,15,20,18,12,5,21,30,30,24,15,6,28,42,45,40,30,

%T 18,7,36,56,63,60,50,36,21,8,45,72,84,84,75,60,42,24,9,55,90,108,112,

%U 105,90,70,48,27,10,66,110,135

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

%C The triangle can be constructed multiplying the triangle A(n,k)=n-k+1 (if 1<=k<=n, else 0) by the triangle B(n,k) =k (if 1<=k<=n, else 0).

%C Swapping the two triangles of this matrix product would generate A104634.

%H G. C. Greubel, <a href="/A104633/b104633.txt">Rows n=1..100 of triangle, flattened</a>

%H Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Falcao/falcao5.html">Intrinsic Properties of a Non-Symmetric Number Triangle</a>, J. Int. Seq., Vol. 26 (2023), Article 23.4.8.

%F G.f.: x*y/((1 - x)^3*(1 - x*y)^2). - _Stefano Spezia_, May 22 2023

%e First few rows of the triangle:

%e 1;

%e 3, 2;

%e 6, 6, 3;

%e 10, 12, 9, 4;

%e 15, 20, 18, 12, 5;

%e 21, 30, 30, 24, 15, 6;

%e 28, 42, 45, 40, 30, 18, 7;

%e 36, 56, 63, 60, 50, 36, 21, 8;

%e ...

%e e.g. Col. 3 = 3 * (1, 3, 6, 10, 15...) = 3, 9, 18, 30, 45...

%p A104633 := proc(n,k) k*(k-n-1)*(k-n-2)/2 ; end proc:

%p seq(seq(A104633(n,k),k=1..n),n=1..16) ; # _R. J. Mathar_, Mar 03 2011

%t Table[k*(k-n-1)*(k-n-2)/2, {n, 1, 20}, {k, 1, n}] // Flatten (* _G. C. Greubel_, Aug 12 2018 *)

%o (PARI) for(n=1,20, for(k=1,n, print1(k*(k-n-1)*(k-n-2)/2, ", "))) \\ _G. C. Greubel_, Aug 12 2018

%o (Magma) [[k*(k-n-1)*(k-n-2)/2: k in [1..n]]: n in [1..20]]; // _G. C. Greubel_, Aug 12 2018

%Y Cf. A062707, A158824, A104634, A001296, A000332 (row sums).

%K nonn,tabl,easy

%O 1,2

%A _Gary W. Adamson_, Mar 18 2005