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

0,5

Comments

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

Links

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

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)

Example

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

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 *)

Prog

(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
(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

Crossrefs

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

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

Sequence in context: A337712 A256184 A120855 * A160001 A339549 A179750

Adjacent sequences: A193734 A193735 A193736 * A193738 A193739 A193740

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 04 2011

Status

approved