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A309557 Number triangle with T(n,k) = 2 + 3*n - 2*k + 2*k*(n-k) for n >= 0, 0 <= k <= n.

%I A309557 #18 Sep 23 2019 15:33:34
%S A309557 2,5,3,8,8,4,11,13,11,5,14,18,18,14,6,17,23,25,23,17,7,20,28,32,32,28,
%T A309557 20,8,23,33,39,41,39,33,23,9,26,38,46,50,50,46,38,26,10,29,43,53,59,
%U A309557 61,59,53,43,29,11,32,48,60,68,72,72,68,60,48,32,12,35,53,67,77,83,85,83,77,67,53,35,13
%N A309557 Number triangle with T(n,k) = 2 + 3*n - 2*k + 2*k*(n-k) for n >= 0, 0 <= k <= n.
%C A309557 The rascal triangle (A077028) can be generated by South = (East*West+1)/North or South = East+West+1-North; this triangle can be generated by South = (East*West+1)/North, South = East+West+2-North.
%H A309557 Philip K Hotchkiss, <a href="https://arxiv.org/abs/1907.11159">Generalized Rascal Triangles</a>, arXiv:1907.11159 [math.HO], 2019.
%F A309557 By rows: a(n,k) = 2 + 3*n - 2*k + 2*k*(n-k) n >= 0, 0 <= k <= n.
%F A309557 By antidiagonals: T(r,k) = 2 + 3*k + r + 2*r*k, r,k >= 0.
%F A309557 G.f.: (x*(-1-4*y+y^2)-2*(1-4*y+y^2))/((-1+x)^2*(-1+y)^3). - _Stefano Spezia_, Sep 08 2019
%e A309557 For row n=3: a(3,0)=11, a(3,1)=13, a(3,2)=11, a(3,3)=5, ...
%e A309557 For antidiagonal r=2: T(2,0)=4, T(2,1)=11, T(2,2)=18, ...
%e A309557 Triangle T begins:
%e A309557               2
%e A309557             5   3
%e A309557           8   8   4
%e A309557         11  13  11  5
%e A309557       14  18  18  14  6
%e A309557     17  23  25  23  17  7
%e A309557   20  28  32  32  28  20  8
%e A309557 23  33  39  41  39  33  23  9
%e A309557              ...
%p A309557 :=proc(n,k)
%p A309557    if n<0 or k<0 or k>n then
%p A309557        0;
%p A309557    else
%p A309557        2+3*n -2*k +2*k*(n-k);
%p A309557    end if;
%t A309557 T[n_,k_]:=2+3*n-2*k+2*k*(n-k); Table[T[n,k], {n,0,11}, {k,0,n}] // Flatten
%t A309557 f[n_] := Table[SeriesCoefficient[(x*(-1-4*y+y^2)-2*(1-4*y+y^2))/((-1+x)^2*(-1+y)^3), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n}]; Flatten[Array[f, 12,0]] (* _Stefano Spezia_, Sep 08 2019 *)
%Y A309557 Cf. A077028, A309555, A309559.
%K A309557 nonn,tabl
%O A309557 0,1
%A A309557 _Philip K Hotchkiss_, Aug 07 2019