%I #28 Dec 02 2023 09:20:28

%S 1,2,1,10,9,2,50,65,28,4,250,425,270,76,8,1250,2625,2200,920,192,16,

%T 6250,15625,16250,9000,2800,464,32,31250,90625,112500,77500,32000,

%U 7920,1088,64,156250,515625,743750,612500,315000,103600,21280,2496,128

%N Mirror of the triangle A193726.

%C This triangle is obtained by reversing the rows of the triangle A193726.

%C Triangle T(n,k), read by rows, given by (2,3,0,0,0,0,0,0,0,...) DELTA (1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 05 2011

%H G. C. Greubel, <a href="/A193727/b193727.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n,k) = A193726(n,n-k).

%F T(n,k) = 2*T(n-1,k-1) + 5*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.: (1-3*x-x*y)/(1-5*x-2*x*y). - _R. J. Mathar_, Aug 11 2015

%F From _G. C. Greubel_, Dec 02 2023: (Start)

%F T(n, 0) = A020699(n).

%F T(n, 1) = A081040(n-1).

%F T(n, n) = A011782(n).

%F Sum_{k=0..n} T(n, k) = A169634(n-1) + (4/7)*[n=0].

%F Sum_{k=0..n} (-1)^k * T(n, k) = A133494(n).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = 2*A015535(n) + A015535(n-1) + (1/2)*[n=0].

%F Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = 2*A107839(n-1) - A107839(n-2) + (1/2)*[n=0]. (End)

%e First six rows:

%e 1;

%e 2, 1;

%e 10, 9, 2;

%e 50, 65, 28, 4;

%e 250, 425, 270, 76, 8;

%e 1250, 2625, 2200, 920, 192; 16;

%t (* First program *)

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

%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}]] (* A193726 *)

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

%t Flatten[Table[g[n], {n, -1, z}]] (* A193727 *)

%t (* Second program *)

%t T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 5*T[n-1, k] + 2*T[n-1, k-1]]];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 02 2023 *)

%o (Magma)

%o function T(n, k) // T = A193727

%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 5*T(n-1, k) + 2*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_, Dec 02 2023

%o (SageMath)

%o def T(n, k): # T = A193727

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

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

%o else: return 5*T(n-1, k) + 2*T(n-1, k-1)

%o flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Dec 02 2023

%Y Cf. A011782, A015535, A020699, A081040, A084938.

%Y Cf. A107839, A133494, A169634, A193722, A193726.

%K nonn,tabl

%O 0,2

%A _Clark Kimberling_, Aug 04 2011