A193728 - OEIS

%I #23 Nov 28 2023 16:24:26

%S 1,2,1,8,10,3,32,64,42,9,128,352,360,162,27,512,1792,2496,1728,594,81,

%T 2048,8704,15360,14400,7560,2106,243,8192,40960,87552,103680,73440,

%U 31104,7290,729,32768,188416,473088,677376,604800,344736,122472,24786,2187

%N Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (x+2)^n and q(n,x) = (2*x+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 (2,2,0,0,0,0,0,0,0,...) 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="/A193728/b193728.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n,k) = 3*T(n-1,k-1) + 4*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-2*x*y)/(1-4*x-3*x*y). - _R. J. Mathar_, Aug 11 2015

%F From _G. C. Greubel_, Nov 28 2023: (Start)

%F T(n, n-k) = A193729(n, k).

%F T(n, 0) = A081294(n).

%F T(n, n-1) = 2*A081038(n-1).

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

%F Sum_{k=0..n} T(n, k) = (1/7)*(4*[n=0] + 3*A000420(n)).

%F Sum_{k=0..n} (-1)^k * T(n, k) = A000012(n).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = (5*b(n) + 4*b(n-1))/14 + (2/3)*[n=0].

%F Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A060816(n),

%F where b(n) = (2 + sqrt(7))^n + (2 - sqrt(7))^n. (End)

%e First six rows:

%e     1;

%e     2,    1;

%e     8,   10,    3;

%e    32,   64,   42,    9;

%e   128,  352,  360,  162,  27;

%e   512, 1792, 2496, 1728, 594, 81;

%t (* First program *)

%t z = 8; a = 1; b = 2; 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}]]  (* A193728 *)

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

%t Flatten[Table[g[n], {n, -1, z}]]   (* A193729 *)

%t (* Second program *)

%t T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 4*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 28 2023 *)

%o (Magma)

%o function T(n, k) // T = A193728

%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 4*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 28 2023

%o (SageMath)

%o def T(n, k): # T = A193728

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

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

%o     else: return 4*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 28 2023

%Y Cf. A000012, A000420, A060816, A081038, A081294.

%Y Cf. A084938, A133494, A193722, A193729.

%K nonn,tabl

%O 0,2

%A _Clark Kimberling_, Aug 04 2011