A165408 An aerated Catalan triangle.

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 0, 0, 3, 0, 1, 0, 0, 2, 0, 4, 0, 1, 0, 0, 0, 5, 0, 5, 0, 1, 0, 0, 0, 0, 9, 0, 6, 0, 1, 0, 0, 0, 5, 0, 14, 0, 7, 0, 1, 0, 0, 0, 0, 14, 0, 20, 0, 8, 0, 1, 0, 0, 0, 0, 0, 28, 0, 27, 0, 9, 0, 1, 0, 0, 0, 0, 14, 0, 48, 0, 35, 0, 10, 0, 1, 0, 0, 0, 0, 0, 42, 0, 75, 0, 44, 0, 11, 0, 1

Offset: 0

Author

Paul Barry, Sep 17 2009

Keywords

• nonn
• tabl
• easy

Comments

Aeration of A120730. Row sums are A165407.

T(n,k) is the number of lattice paths from (0,0) to (k,(n-k)/2) that do not go above the diagonal x=y using steps in {(1,0), (0,1)}. - Alois P. Heinz, Sep 20 2022

Examples

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 1, 0, 1;
  0, 0, 2, 0,  1;
  0, 0, 0, 3,  0,  1;
  0, 0, 2, 0,  4,  0,  1;
  0, 0, 0, 5,  0,  5,  0,  1;
  0, 0, 0, 0,  9,  0,  6,  0,  1;
  0, 0, 0, 5,  0, 14,  0,  7,  0, 1;
  0, 0, 0, 0, 14,  0, 20,  0,  8, 0,  1;
  0, 0, 0, 0,  0, 28,  0, 27,  0, 9,  0, 1;
  0, 0, 0, 0, 14,  0, 48,  0, 35, 0, 10, 0, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Cf. A000108, A001405, A001477, A105523, A120730, A165407 (row sums), A165409.

Cf. A174687, A236194, A262394.

Programs

Magma

A165408:= func< n,k | n le 3*k select Binomial(Floor((n+k)/2), k)*((3*k-n)/2 +1)*(1+(-1)^(n-k))/(2*(k+1)) else 0 >;
[A165408(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Nov 09 2022

Maple

b:= proc(x, y) option remember; `if`(y<=x, `if`(x=0, 1,
      b(x-1, y)+`if`(y>0, b(x, y-1), 0)), 0)
    end:
T:= (n, k)-> `if`((n-k)::even, b(k, (n-k)/2), 0):
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Sep 20 2022

Mathematica

b[x_, y_]:= b[x, y]= If[y<=x, If[x==0, 1, b[x-1, y] + If[y>0, b[x, y-1], 0]], 0];
T[n_, k_]:= If[EvenQ[n-k], b[k, (n-k)/2], 0];
Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* Jean-François Alcover, Oct 08 2022, after Alois P. Heinz *)

SageMath

def A165408(n,k): return 0 if (n>3*k) else binomial(int((n+k)/2), k)*((3*k-n+2)/2 )*(1+(-1)^(n-k))/(2*(k+1))
flatten([[A165408(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Nov 09 2022

Formula

T(n,k) = if(n<=3k, C((n+k)/2, k)*((3*k-n)/2 + 1)*(1 + (-1)^(n-k))/(2*(k+1)), 0).

G.f.: 1/(1-x*y-x^3*y/(1-x^3*y/(1-x^3*y/(1-x^3*y/(1-... (continued fraction).

Sum_{k=0..n} T(n, k) = A165407(n).

From G. C. Greubel, Nov 09 2022: (Start)

Sum_{k=0..floor(n/2)} T(n-k, k) = (1+(-1)^n)*A001405(n/2)/2.

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1+(-1)^n)*A105523(n/2)/2.

Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A165407(n).

Sum_{k=0..n} 2^k*T(n, k) = A165409(n).

T(n, n-2) = A001477(n-2), n >= 2.

T(2*n, n) = (1+(-1)^n)*A174687(n/2)/2.

T(2*n, n+1) = (1-(-1)^n)*A262394(n/2)/2.

T(2*n, n-1) = (1+(-1)^n)*A236194(n/2)/2

T(3*n-2, n) = A000108(n), n >= 1. (End)