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Number triangle with T(n,k) = 2 + 3*n - 2*k + 2*k*(n-k) for n >= 0, 0 <= k <= n.
4

%I #18 Sep 23 2019 15:33:34

%S 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 20,8,23,33,39,41,39,33,23,9,26,38,46,50,50,46,38,26,10,29,43,53,59,

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

%C 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 Philip K Hotchkiss, <a href="https://arxiv.org/abs/1907.11159">Generalized Rascal Triangles</a>, arXiv:1907.11159 [math.HO], 2019.

%F By rows: a(n,k) = 2 + 3*n - 2*k + 2*k*(n-k) n >= 0, 0 <= k <= n.

%F By antidiagonals: T(r,k) = 2 + 3*k + r + 2*r*k, r,k >= 0.

%F 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 For row n=3: a(3,0)=11, a(3,1)=13, a(3,2)=11, a(3,3)=5, ...

%e For antidiagonal r=2: T(2,0)=4, T(2,1)=11, T(2,2)=18, ...

%e Triangle T begins:

%e 2

%e 5 3

%e 8 8 4

%e 11 13 11 5

%e 14 18 18 14 6

%e 17 23 25 23 17 7

%e 20 28 32 32 28 20 8

%e 23 33 39 41 39 33 23 9

%e ...

%p :=proc(n,k)

%p if n<0 or k<0 or k>n then

%p 0;

%p else

%p 2+3*n -2*k +2*k*(n-k);

%p end if;

%t T[n_,k_]:=2+3*n-2*k+2*k*(n-k); Table[T[n,k], {n,0,11}, {k,0,n}] // Flatten

%t 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 Cf. A077028, A309555, A309559.

%K nonn,tabl

%O 0,1

%A _Philip K Hotchkiss_, Aug 07 2019