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%I #8 Apr 01 2021 14:53:56
%S 1,4,1,10,3,2,20,6,6,3,35,10,12,9,4,56,15,20,18,12,5,84,21,30,30,24,
%T 15,6,120,28,42,45,40,30,18,7,165,36,56,63,60,50,36,21,8,220,45,72,84,
%U 84,75,60,42,24,9,286,55,90,108,112,105,90,70,48,27,10,364,66,110,135,144,140,126,105,80,54,30,11
%N Triangle T(n,k) = A000292(n) if k = 1 otherwise (k-1)*(n-k+1)*(n-k+2)/2, read by rows.
%C The triangle can also be defined by multiplying the triangles A(n,k)=1 and A158823(n,k), that is, this here are the partial column sums of A158823.
%H G. C. Greubel, <a href="/A158824/b158824.txt">Rows n = 1..50 of the triangle, flattened</a>
%F T(n,k) = binomial(n+2,3) if k = 1 otherwise (k-1)*binomial(n-k+2, 2).
%F Sum_{k=1..n} T(n, k) = binomial(n+3, 4) = A000332(n+3). - _G. C. Greubel_, Apr 01 2021
%e First few rows of the triangle are:
%e 1;
%e 4, 1;
%e 10, 3, 2;
%e 20, 6, 6, 3;
%e 35, 10, 12, 9, 4;
%e 56, 15, 20, 18, 12, 5;
%e 84, 21, 30, 30, 24, 15, 6;
%e 120, 28, 42, 45, 40, 30, 18, 7;
%e 165, 36, 56, 63, 60, 50, 36, 21, 8;
%e 220, 45, 72, 84, 84, 75, 60, 42, 24, 9;
%e 286, 55, 90, 108, 112, 105, 90, 70, 48, 27, 10;
%e 364, 66, 110, 135, 144, 140, 126, 105, 80, 54, 30, 11;
%e 455, 78, 132, 165, 180, 180, 168, 147, 120, 90, 60, 33, 12;
%e ...
%t T[n_, k_]:= If[k==1, Binomial[n+2, 3], (k-1)*Binomial[n-k+2, 2]];
%t Table[T[n, k], {n, 12}, {k, n}]//Flatten (* _G. C. Greubel_, Apr 01 2021 *)
%o (Magma) A158824:= func< n,k | k eq 1 select Binomial(n+2,3) else (k-1)*Binomial(n-k+2,2) >; [A158824(n, k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Apr 01 2021
%o (Sage)
%o def A158824(n,k): return binomial(n+2,3) if k==1 else (k-1)*binomial(n-k+2,2)
%o flatten([[A158824(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Apr 01 2021
%Y Cf. A062707, A104633, A158823.
%Y Row sums: A000332.
%K nonn,tabl,easy
%O 1,2
%A _Gary W. Adamson_ & _Roger L. Bagula_, Mar 28 2009
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