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

0,3

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

Links

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

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)

Example

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

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 *)

Prog

(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
(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

Crossrefs

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

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

Cf. A193722, A193730.

Sequence in context: A110142 A158928 A198369 * A193975 A224665 A098514

Adjacent sequences: A193728 A193729 A193730 * A193732 A193733 A193734

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 04 2011

Status

approved