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A193846 Triangular array: the fission of ((x+2)^n) by ((x+1)^n).

%I A193846 #16 Feb 18 2024 14:41:34
%S A193846 2,4,8,8,28,26,16,80,136,80,32,208,512,568,242,64,512,1648,2672,2188,
%T A193846 728,128,1216,4832,10288,12392,8020,2186,256,2816,13312,35072,55648,
%U A193846 53216,28432,6560,512,6400,35072,110080,216512,273376,216512,98416
%N A193846 Triangular array: the fission of ((x+2)^n) by ((x+1)^n).
%C A193846 See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.
%H A193846 Andrew Howroyd, <a href="/A193846/b193846.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%F A193846 G.f.: A(x,y) = 2/(1 - 2*(1 + 2*y)*x + y*(2 + 3*y)*x^2). - _Andrew Howroyd_, Feb 18 2024
%e A193846 First six rows:
%e A193846 2
%e A193846 4....8
%e A193846 8....28....26
%e A193846 16...80....136....80
%e A193846 32...208...512....568....242
%e A193846 64...512...1648...2672...2188...728
%t A193846 p[n_, x_] := (x + 2)^n;
%t A193846 q[n_, x_] := (x + 1)^n
%t A193846 p1[n_, k_] := Coefficient[p[n, x], x^k];
%t A193846 p1[n_, 0] := p[n, x] /. x -> 0;
%t A193846 d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t A193846 h[n_] := CoefficientList[d[n, x], {x}]
%t A193846 TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t A193846 Flatten[Table[Reverse[h[n]], {n, -1, z}]]   (* A193846 *)
%t A193846 TableForm[Table[h[n], {n, 0, z}]]
%t A193846 Flatten[Table[h[n], {n, -1, z}]]  (* A193847 *)
%t A193846 TableForm[Table[Reverse[h[n]/2], {n, 0, z}]]
%t A193846 Flatten[Table[Reverse[h[n]]/2, {n, -1, z}]] (* A193848 *)
%t A193846 TableForm[Table[h[n]/2, {n, 0, z}]]
%t A193846 Flatten[Table[h[n]/2, {n, -1, z}]]  (* A193849 *)
%o A193846 (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 A193846 { my(A=T(10)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Feb 18 2024
%Y A193846 Cf. A193842, A193847, A193848.
%K A193846 nonn,tabl
%O A193846 0,1
%A A193846 _Clark Kimberling_, Aug 07 2011