%I #16 Feb 18 2024 14:41:34

%S 2,4,8,8,28,26,16,80,136,80,32,208,512,568,242,64,512,1648,2672,2188,

%T 728,128,1216,4832,10288,12392,8020,2186,256,2816,13312,35072,55648,

%U 53216,28432,6560,512,6400,35072,110080,216512,273376,216512,98416

%N Triangular array: the fission of ((x+2)^n) by ((x+1)^n).

%C See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.

%H Andrew Howroyd, <a href="/A193846/b193846.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

%F G.f.: A(x,y) = 2/(1 - 2*(1 + 2*y)*x + y*(2 + 3*y)*x^2). - _Andrew Howroyd_, Feb 18 2024

%e First six rows:

%e 2

%e 4....8

%e 8....28....26

%e 16...80....136....80

%e 32...208...512....568....242

%e 64...512...1648...2672...2188...728

%t p[n_, x_] := (x + 2)^n;

%t q[n_, x_] := (x + 1)^n

%t p1[n_, k_] := Coefficient[p[n, x], x^k];

%t p1[n_, 0] := p[n, x] /. x -> 0;

%t d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]

%t h[n_] := CoefficientList[d[n, x], {x}]

%t TableForm[Table[Reverse[h[n]], {n, 0, z}]]

%t Flatten[Table[Reverse[h[n]], {n, -1, z}]] (* A193846 *)

%t TableForm[Table[h[n], {n, 0, z}]]

%t Flatten[Table[h[n], {n, -1, z}]] (* A193847 *)

%t TableForm[Table[Reverse[h[n]/2], {n, 0, z}]]

%t Flatten[Table[Reverse[h[n]]/2, {n, -1, z}]] (* A193848 *)

%t TableForm[Table[h[n]/2, {n, 0, z}]]

%t Flatten[Table[h[n]/2, {n, -1, z}]] (* A193849 *)

%o (PARI) T(n)={[2*Vecrev(p) | p<-Vec(1/(1 - 2*(1 + 2*y)*x + y*(2 + 3*y)*x^2) + O(x*x^n))]}

%o { my(A=T(10)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Feb 18 2024

%Y Cf. A193842, A193847, A193848.

%K nonn,tabl

%O 0,1

%A _Clark Kimberling_, Aug 07 2011