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A193858 Triangular array: the fission of ((x+1)^n) by ((2x+1)^n).

%I A193858 #22 Feb 18 2024 13:39:11
%S A193858 1,2,3,4,10,7,8,28,34,15,16,72,124,98,31,32,176,392,444,258,63,64,416,
%T A193858 1136,1672,1404,642,127,128,960,3104,5616,6152,4092,1538,255,256,2176,
%U A193858 8128,17440,23536,20488,11260,3586,511,512,4864,20608,51136,81952
%N A193858 Triangular array:  the fission of ((x+1)^n) by ((2x+1)^n).
%C A193858 See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.
%H A193858 Andrew Howroyd, <a href="/A193858/b193858.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%F A193858 From _Andrew Howroyd_, Feb 18 2024: (Start)
%F A193858 T(n,k) = Sum_{j=0..k} 2^(n-j) * binomial(n-j,k-j).
%F A193858 G.f.: A(x,y) = 1/(1 - (2 + 3*y)*x + 2*y*(1 + y)*x^2). (End)
%e A193858 First six rows:
%e A193858    1
%e A193858    2     3
%e A193858    4    10     7
%e A193858    8    28    34    15
%e A193858   16    72   124    98    31
%e A193858   32   176   392   444   258   63
%p A193858 # The function 'fission' is defined in A193842.
%p A193858 A193858_row := n -> fission((n,x) -> (x+1)^n, (n,x) -> (2*x+1)^n, n);
%p A193858 for n from 0 to 5 do A193858_row(n) od; # _Peter Luschny_, Jul 23 2014
%t A193858 z = 10;
%t A193858 p[n_, x_] := (x + 1)^n;
%t A193858 q[n_, x_] := (2 x + 1)^n;
%t A193858 p1[n_, k_] := Coefficient[p[n, x], x^k];
%t A193858 p1[n_, 0] := p[n, x] /. x -> 0;
%t A193858 d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t A193858 h[n_] := CoefficientList[d[n, x], {x}]
%t A193858 TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t A193858 Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* this sequence *)
%t A193858 TableForm[Table[h[n], {n, 0, z}]]
%t A193858 Flatten[Table[h[n], {n, -1, z}]]   (* A193859 *)
%o A193858 (Sage) # uses[fission from A193842]
%o A193858 A193858_row = lambda k: fission(lambda n,x: (x+1)^n, lambda n,x: (2*x+1)^n, k)
%o A193858 for n in range(7): A193858_row(n) # _Peter Luschny_, Jul 23 2014
%o A193858 (PARI) T(n,k)={sum(j=0, k, 2^(n-j) * binomial(n-j, k-j))} \\ _Andrew Howroyd_, Feb 18 2024
%Y A193858 Cf. A193842, A193859.
%K A193858 nonn,tabl
%O A193858 0,2
%A A193858 _Clark Kimberling_, Aug 07 2011