A193731 Mirror of the triangle A193730.

1, 1, 2, 3, 8, 4, 9, 30, 28, 8, 27, 108, 144, 80, 16, 81, 378, 648, 528, 208, 32, 243, 1296, 2700, 2880, 1680, 512, 64, 729, 4374, 10692, 14040, 10800, 4896, 1216, 128, 2187, 14580, 40824, 63504, 60480, 36288, 13440, 2816, 256, 6561, 48114, 151632, 272160, 308448, 229824, 112896, 35328, 6400, 512

Offset: 0

Author

Clark Kimberling, Aug 04 2011

Keywords

• nonn
• tabl

Comments

A193731 is obtained by reversing the rows of the triangle A193730.

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

Examples

			First six rows:
   1;
   1,   2;
   3,   8,   4;
   9,  30,  28,   8;
  27, 108, 144,  80,  16;
  81, 378, 648, 528, 208, 32;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000012, A000079, A005053, A006234, A007483, A080420, A080421.

Cf. A080422, A080423, A084938, A130129, A133494, A153881.

Cf. A193722, A193730.

Programs

Magma

function T(n, k) // T = A193731
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return k+1;
  else return 3*T(n-1, k) + 2*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2023

Mathematica

(* First program *)
z = 8; a = 2; b = 1; c = 2; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193730 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]     (* A193731 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, k+1, 3*T[n-1, k] + 2*T[n -1, k-1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2023 *)

SageMath

def T(n, k): # T = A193731
    if (k<0 or k>n): return 0
    elif (n<2): return k+1
    else: return 3*T(n-1, k) + 2*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 20 2023

Formula

T(n,k) = A193730(n,n-k).

T(n,k) = 2*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

G.f.: (1-2*x)/(1-3*x-2*x*y). - R. J. Mathar, Aug 11 2015

From G. C. Greubel, Nov 20 2023: (Start)

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

T(n, 1) = 2*A006234(n+2).

T(n, 2) = 4*A080420(n-2).

T(n, 3) = 8*A080421(n-3).

T(n, 4) = 16*A080422(n-4).

T(n, 5) = 32*A080423(n-5).

T(n, n) = A000079(n).

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

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

Sum_{k=0..n} (-1)^k * T(n, k) = A153881(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = A007483(n-1).

Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A000012(n). (End)