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A094415 Triangle T read by rows: dot product * .

%I A094415 #10 Sep 08 2022 08:45:13
%S A094415 1,4,5,10,13,13,20,26,28,26,35,45,50,50,45,56,71,80,83,80,71,84,105,
%T A094415 119,126,126,119,105,120,148,168,180,184,180,168,148,165,201,228,246,
%U A094415 255,255,246,228,201,220,265,300,325,340,345,340,325,300,265,286,341
%N A094415 Triangle T read by rows: dot product <r,r-1,...,1> * <s+1,s+2,...,r,1,2,...,s>.
%H A094415 G. C. Greubel, <a href="/A094415/b094415.txt">Rows n = 0..100 of triangle, flattened</a>
%F A094415 T(n, k) = n*(n^2 + 3*n*(1+k) + 2 - 3*k^2)/6 for n >= 0, 0 <= k <= n.
%e A094415 Triangle begins as:
%e A094415    1;
%e A094415    4,  5;
%e A094415   10, 13, 13;
%e A094415   20, 26, 28, 26;
%e A094415   35, 45, 50, 50, 45;
%e A094415   56, 71, 80, 83, 80, 71;
%p A094415 seq(seq( (n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6 , k=0..n), n=0..12); # _G. C. Greubel_, Oct 30 2019
%t A094415 Table[(n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6, {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Oct 30 2019 *)
%o A094415 (PARI) T(n,k) = (n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6;
%o A094415 for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Oct 30 2019
%o A094415 (Magma) [(n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6: k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 30 2019
%o A094415 (Sage) [[(n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6 for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Oct 30 2019
%o A094415 (GAP) Flat(List([0..12], n-> List([0..n], k-> (n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6 ))); # _G. C. Greubel_, Oct 30 2019
%Y A094415 Columns 0-6 are A000292, A008778, A026054, A026057, A026060, A026063, A026066.
%Y A094415 Half-diagonal is A050410.
%Y A094415 Row sums are A000537.
%Y A094415 See also A094414, A088003.
%K A094415 nonn,tabl
%O A094415 0,2
%A A094415 _Ralf Stephan_, May 02 2004