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Mirror of the triangle A193728.
3

%I #27 Nov 28 2023 15:56:55

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

%T 243,2106,7560,14400,15360,8704,2048,729,7290,31104,73440,103680,

%U 87552,40960,8192,2187,24786,122472,344736,604800,677376,473088,188416,32768

%N Mirror of the triangle A193728.

%C T(n, k) is obtained by reversing the rows of the triangle A193728.

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

%H G. C. Greubel, <a href="/A193729/b193729.txt">Rows n = 0..50 of the triangle, flattened</a>

%F Let w(n,k) be the triangle of A193728, then the triangle in this sequence is given by w(n,n-k).

%F T(n,k) = 4*T(n-1,k-1) + 3*T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - _Philippe Deléham_, Oct 05 2011

%F G.f.: (1-2*x-2*x*y)/(1-3*x-4*x*y). - _R. J. Mathar_, Aug 11 2015

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

%F T(n, 0) = A133494(n).

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

%F T(n, n) = A081294(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) = A033999(n).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*[n=0] + A108981(n-1).

%F Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = (1/2)*[n=0] + A247560(n-1).

%F (End)

%e First six rows:

%e 1;

%e 1, 2;

%e 3, 10, 8;

%e 9, 42, 64, 32;

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

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

%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, k+1, 3*T[n-1,k] + 4*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 = A193729

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

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

%o else return 3*T(n-1, k) + 4*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 = A193729

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

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

%o else: return 3*T(n-1, k) + 4*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. A000420, A033999, A081038, A081294, A084938.

%Y Cf. A108981, A133494, A193722, A193728, A247560.

%K nonn,tabl

%O 0,3

%A _Clark Kimberling_, Aug 04 2011