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%I #25 Apr 23 2020 22:31:08
%S 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 22,8,14,19,23,26,28,29,29,9,16,22,27,31,34,36,37,37,10,18,25,31,36,
%U 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 Triangle read by rows: T(n,k) = 1 + n + k^2/2 - k/2 + k*(n-k), n >= 0, 0 <= k <= n.
%C 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 Philip K Hotchkiss, <a href="https://arxiv.org/abs/1907.11159">Generalized Rascal Triangles</a>, arXiv:1907.11159 [math.HO], 2019.
%F G.f.: (-1+(3-2*x)*y+(-1+x)*y^2)/((-1+x)^2*(-1+y)^3). - _Stefano Spezia_, Sep 08 2019
%e For row n=3: T(3,0)=4, T(3,1)=6, T(3,2)=6, T(3,3)=7.
%e Triangle T begins:
%e 1
%e 2 2
%e 3 4 4
%e 4 6 7 7
%e 5 8 10 11 11
%e 6 10 13 15 16 16
%e 7 12 16 19 21 22 22
%e 8 14 19 23 26 28 29 29
%e 9 16 22 27 31 34 36 37 37
%e ...
%p T := proc(n, k)
%p if n<0 or k<0 or k>n then
%p 0;
%p else
%p 1+n+(1/2)*k^2-(1/2)k +k*(n-k);
%p end if;
%t 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 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 Cf. A077028, A309555, A309557.
%K nonn,tabl
%O 0,2
%A _Philip K Hotchkiss_, Aug 07 2019
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