A193737 Mirror of the triangle A193736.

1, 1, 1, 1, 2, 1, 2, 4, 3, 1, 3, 8, 8, 4, 1, 5, 15, 19, 13, 5, 1, 8, 28, 42, 36, 19, 6, 1, 13, 51, 89, 91, 60, 26, 7, 1, 21, 92, 182, 216, 170, 92, 34, 8, 1, 34, 164, 363, 489, 446, 288, 133, 43, 9, 1, 55, 290, 709, 1068, 1105, 826, 455, 184, 53, 10, 1, 89, 509, 1362, 2266, 2619, 2219, 1414, 682, 246, 64, 11, 1

Offset: 0

Author

Clark Kimberling, Aug 04 2011

Keywords

• nonn
• tabl

Comments

This triangle is obtained by reversing the rows of the triangle A193736.

Examples

			First six rows:
  1;
  1,  1;
  1,  2,  1;
  2,  4,  3,  1;
  3,  8,  8,  4,  1;
  5, 15, 19, 13,  5,  1;
		

Links

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

Crossrefs

Cf. A000007, A011782 (diagonal sums), A019590, A052542 (row sums).

Cf. A034856, A193736, A321620, A324969, A330793.

Programs

Magma

function T(n,k) // T = A193737
  if k lt 0 or n lt 0 then return 0;
  elif n lt 3 then return Binomial(n,k);
  else return T(n - 1, k) + T(n - 1, k - 1) + T(n - 2, k);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 24 2023

Mathematica

(* First program *)
z=20;
p[0, x_]:= 1;
p[n_, x_]:= Fibonacci[n+1, x] /; n > 0
q[n_, x_]:= (x + 1)^n;
t[n_, k_]:= Coefficient[p[n, x], x^(n-k)];
t[n_, n_]:= p[n, x] /. x -> 0;
w[n_, x_]:= Sum[t[n, k]*q[n-k+1, 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}]] (* A193736 *)
TableForm[Table[g[n], {n,-1,z}]]
Flatten[Table[g[n], {n,-1,z}]]          (* A193737 *)
(* Additional programs *)
(* Function RiordanSquare defined in A321620. *)
RiordanSquare[1 + 1/(1 - x - x^2), 11]//Flatten  (* Peter Luschny, Feb 27 2021 *)
T[n_, k_]:= T[n, k]= If[n<3, Binomial[n, k], T[n-1,k] + T[n-1,k-1] + T[n-2,k]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 24 2023 *)

SageMath

def T(n,k): # T = A193737
    if (n<3): return binomial(n,k)
    else: return T(n-1,k) +T(n-1,k-1) +T(n-2,k)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 24 2023

Formula

Write w(n,k) for the triangle at A193736. This is then given by w(n,n-k).

T(0,0) = T(1,0) = T(1,1) = T(2,0) = 1; T(n,k) = 0 if k<0 or k>n; T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k). - Philippe Deléham, Feb 13 2020

From G. C. Greubel, Oct 24 2023: (Start)

T(n, 0) = Fibonacci(n) + [n=0] = A324969(n+1).

T(n, n-1) = n, for n >= 1.

T(n, n-2) = A034856(n-1), for n >= 2.

T(2*n, n) = A330793(n).

Sum_{k=0..n} T(n,k) = A052542(n).

Sum_{k=0..n} (-1)^k * T(n,k) = A000007(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = A011782(n).

Sum_{k=0..floor(n/2)} (-1)^k * T(n-k,k) = A019590(n). (End)