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A104633 Triangle T(n,k) = k*(k-n-1)*(k-n-2)/2 read by rows, 1<=k<=n.

%I A104633 #26 Apr 04 2025 15:38:32
%S A104633 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 A104633 18,7,36,56,63,60,50,36,21,8,45,72,84,84,75,60,42,24,9,55,90,108,112,
%U A104633 105,90,70,48,27,10,66,110,135
%N A104633 Triangle T(n,k) = k*(k-n-1)*(k-n-2)/2 read by rows, 1<=k<=n.
%C A104633 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 A104633 Swapping the two triangles of this matrix product would generate A104634.
%H A104633 G. C. Greubel, <a href="/A104633/b104633.txt">Rows n=1..100 of triangle, flattened</a>
%H A104633 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.
%H A104633 Joaquín Figueroa, Ivan Gonzalez, and Daniel Salinas-Arizmendi, <a href="https://arxiv.org/abs/2503.23134">A Novel Transfer Matrix Framework for Multiple Dirac Delta Potentials</a>, arXiv:2503.23134 [quant-ph], 2025. See pp. 4, 9.
%F A104633 G.f.: x*y/((1 - x)^3*(1 - x*y)^2). - _Stefano Spezia_, May 22 2023
%e A104633 First few rows of the triangle:
%e A104633   1;
%e A104633   3,  2;
%e A104633   6,  6,  3;
%e A104633  10, 12,  9,  4;
%e A104633  15, 20, 18, 12,  5;
%e A104633  21, 30, 30, 24, 15,  6;
%e A104633  28, 42, 45, 40, 30, 18,  7;
%e A104633  36, 56, 63, 60, 50, 36, 21, 8;
%e A104633  ...
%e A104633 e.g. Col. 3 = 3 * (1, 3, 6, 10, 15...) = 3, 9, 18, 30, 45...
%p A104633 A104633 := proc(n,k) k*(k-n-1)*(k-n-2)/2 ; end proc:
%p A104633 seq(seq(A104633(n,k),k=1..n),n=1..16) ; # _R. J. Mathar_, Mar 03 2011
%t A104633 Table[k*(k-n-1)*(k-n-2)/2, {n, 1, 20}, {k, 1, n}] // Flatten (* _G. C. Greubel_, Aug 12 2018 *)
%o A104633 (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 A104633 (Magma) [[k*(k-n-1)*(k-n-2)/2: k in [1..n]]: n in [1..20]]; // _G. C. Greubel_, Aug 12 2018
%Y A104633 Cf. A062707, A158824, A104634, A001296, A000332 (row sums).
%K A104633 nonn,tabl,easy
%O A104633 1,2
%A A104633 _Gary W. Adamson_, Mar 18 2005