A193727 Mirror of the triangle A193726.

1, 2, 1, 10, 9, 2, 50, 65, 28, 4, 250, 425, 270, 76, 8, 1250, 2625, 2200, 920, 192, 16, 6250, 15625, 16250, 9000, 2800, 464, 32, 31250, 90625, 112500, 77500, 32000, 7920, 1088, 64, 156250, 515625, 743750, 612500, 315000, 103600, 21280, 2496, 128

Offset

0,2

Comments

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

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

Links

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

Formula

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

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

G.f.: (1-3*x-x*y)/(1-5*x-2*x*y). - R. J. Mathar, Aug 11 2015

From G. C. Greubel, Dec 02 2023: (Start)

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

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

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

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

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

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

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

Example

First six rows:
     1;
     2,    1;
    10,    9,    2;
    50,   65,   28,   4;
   250,  425,  270,  76,   8;
  1250, 2625, 2200, 920, 192; 16;

Mathematica

(* First program *)
z = 8; a = 1; b = 2; c = 1; d = 2;
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}]]  (* A193726 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193727 *)
(* Second program *)
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]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 02 2023 *)

Prog

(Magma)
function T(n, k) // T = A193727
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return n-k+1;
  else return 5*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, Dec 02 2023
(SageMath)
def T(n, k): # T = A193727
    if (k<0 or k>n): return 0
    elif (n<2): return n-k+1
    else: return 5*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, Dec 02 2023

Crossrefs

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

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

Sequence in context: A127259 A152260 A286781 * A138098 A081098 A244582

Adjacent sequences: A193724 A193725 A193726 * A193728 A193729 A193730

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 04 2011

Status

approved