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Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (2x+1)^n and q(n,x) = (2x+1)^n.
3

%I #24 Nov 21 2023 17:12:06

%S 1,2,1,4,8,3,8,28,30,9,16,80,144,108,27,32,208,528,648,378,81,64,512,

%T 1680,2880,2700,1296,243,128,1216,4896,10800,14040,10692,4374,729,256,

%U 2816,13440,36288,60480,63504,40824,14580,2187,512,6400,35328,112896,229824,308448,272160,151632,48114,6561

%N Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (2x+1)^n and q(n,x) = (2x+1)^n.

%C See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

%C Triangle T(n,k), read by rows, given by (2,0,0,0,0,0,0,0,...) DELTA (1,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 05 2011

%H G. C. Greubel, <a href="/A193730/b193730.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n,k) = 3*T(n-1,k-1) + 2*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - _Philippe Deléham_, Oct 05 2011

%F G.f.: (1-2*x*y)/(1-2*x-3*x*y). - _R. J. Mathar_, Aug 11 2015

%F From _G. C. Greubel_, Nov 20 2023: (Start)

%F T(n, 0) = A000079(n).

%F T(n, 1) = A130129(n-1).

%F T(n, n) = A133494(n).

%F T(n, n-1) = A199923(n).

%F Sum_{k=0..n} T(n, k) = A005053(n).

%F Sum_{k=0..n} (-1)^k * T(n, k) = A165326(n). (End)

%e First six rows:

%e 1;

%e 2, 1;

%e 4, 8, 3;

%e 8, 28, 30, 9;

%e 16, 80, 144, 108, 27;

%e 32, 208, 528, 648, 378, 81;

%t (* First program *)

%t z = 8; a = 2; b = 1; c = 2; d = 1;

%t p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n

%t t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;

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

%t g[n_] := CoefficientList[w[n, x], {x}]

%t TableForm[Table[Reverse[g[n]], {n, -1, z}]]

%t Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193730 *)

%t TableForm[Table[g[n], {n, -1, z}]]

%t Flatten[Table[g[n], {n, -1, z}]] (* A193731 *)

%t (* Second program *)

%t T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 2*T[n-1, k] + 3*T[n-1, k-1]]];

%t Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 20 2023 *)

%o (Magma)

%o function T(n, k) // T = A193730

%o if k lt 0 or k gt n then return 0;

%o elif n lt 2 then return n-k+1;

%o else return 2*T(n-1, k) + 3*T(n-1, k-1);

%o end if;

%o end function;

%o [T(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 20 2023

%o (SageMath)

%o def T(n, k): # T = A193730

%o if (k<0 or k>n): return 0

%o elif (n<2): return n-k+1

%o else: return 2*T(n-1, k) + 3*T(n-1, k-1)

%o flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Nov 20 2023

%Y Cf. A000079, A005053, A084938, A130129, A133494, A165326.

%Y Cf. A193722, A193731, A199923.

%K nonn,tabl

%O 0,2

%A _Clark Kimberling_, Aug 04 2011