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A309559 Triangle read by rows: T(n,k) = 1 + n + k^2/2 - k/2 + k*(n-k), n >= 0, 0 <= k <= n.

%I A309559 #25 Apr 23 2020 22:31:08
%S A309559 1,2,2,3,4,4,4,6,7,7,5,8,10,11,11,6,10,13,15,16,16,7,12,16,19,21,22,
%T A309559 22,8,14,19,23,26,28,29,29,9,16,22,27,31,34,36,37,37,10,18,25,31,36,
%U A309559 40,43,45,46,46,11,20,28,35,41,46,50,53,55,56,56,12,22,31,39,46,52,57,61,64,66,67,67,13,24,34,43,51,58,64,69,73,76,78,79,79
%N A309559 Triangle read by rows: T(n,k) = 1 + n + k^2/2 - k/2 + k*(n-k), n >= 0, 0 <= k <= n.
%C A309559 The rascal triangle (A077028) can be generated by  the rule South = (East*West+1)/North or South = East+West+1-North; this number triangle can also be generated by South = East+West+1-North, but there not by an equation of the form South = (East*West+d)/North.
%H A309559 Philip K Hotchkiss, <a href="https://arxiv.org/abs/1907.11159">Generalized Rascal Triangles</a>, arXiv:1907.11159 [math.HO], 2019.
%F A309559 G.f.: (-1+(3-2*x)*y+(-1+x)*y^2)/((-1+x)^2*(-1+y)^3). - _Stefano Spezia_, Sep 08 2019
%e A309559 For row n=3: T(3,0)=4, T(3,1)=6, T(3,2)=6, T(3,3)=7.
%e A309559 Triangle T begins:
%e A309559                   1
%e A309559                 2   2
%e A309559               3   4   4
%e A309559             4   6   7   7
%e A309559           5   8  10  11  11
%e A309559         6  10  13  15  16  16
%e A309559       7  12  16  19  21  22  22
%e A309559     8  14  19  23  26  28  29  29
%e A309559   9  16  22  27  31  34  36  37  37
%e A309559                  ...
%p A309559 T := proc(n, k)
%p A309559    if n<0 or k<0 or k>n then
%p A309559        0;
%p A309559    else
%p A309559        1+n+(1/2)*k^2-(1/2)k +k*(n-k);
%p A309559    end if;
%t A309559 T[n_,k_]:=1+n+(1/2)*k^2-(1/2)k +k*(n-k); Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten
%t A309559 f[n_] := Table[SeriesCoefficient[(-1+(3-2*x)*y+(-1+x)*y^2)/((-1+x)^2*(-1+y)^3), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n}]; Flatten[Array[f, 13,0]] (* _Stefano Spezia_, Sep 08 2019 *)
%Y A309559 Cf. A077028, A309555, A309557.
%K A309559 nonn,tabl
%O A309559 0,2
%A A309559 _Philip K Hotchkiss_, Aug 07 2019