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A158824 Triangle T(n,k) = A000292(n) if k = 1 otherwise (k-1)*(n-k+1)*(n-k+2)/2, read by rows.

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