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

0,13

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

Links

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

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)

Example

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;
  ...

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 *)

Prog

(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
(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

Crossrefs

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

Cf. A174687, A236194, A262394.

Sequence in context: A187144 A123635 A124304 * A186733 A332042 A171368

Adjacent sequences: A165405 A165406 A165407 * A165409 A165410 A165411

Keyword

nonn,tabl,easy

Author

Paul Barry, Sep 17 2009

Status

approved