A155839 A ratio of two Catalan arrays.

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 3, 0, 1, 0, 4, 7, 6, 0, 1, 0, 8, 18, 16, 10, 0, 1, 0, 16, 45, 51, 30, 15, 0, 1, 0, 32, 110, 152, 115, 50, 21, 0, 1, 0, 64, 264, 436, 396, 225, 77, 28, 0, 1, 0, 128, 624, 1212, 1300, 876, 399, 112, 36, 0, 1

Offset: 0

Author

Paul Barry, Jan 28 2009

Keywords

• easy
• nonn
• tabl

Examples

			Triangle begins
  1;
  0,  1;
  0,  0,   1;
  0,  1,   0,   1;
  0,  2,   3,   0,   1;
  0,  4,   7,   6,   0,  1;
  0,  8,  18,  16,  10,  0,  1;
  0, 16,  45,  51,  30, 15,  0, 1;
  0, 32, 110, 152, 115, 50, 21, 0, 1;
		

Links

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

Crossrefs

Cf. A000108, A033184, A120010 (row sums), A124644.

Programs

Magma

A155839:= func< n,k | (&+[(-1)^(n-j)*Binomial(j+1, n-j)*Binomial(j, k)*Catalan(j-k) : j in [k..n]]) >;
[A155839(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 04 2021

Mathematica

T[n_, k_] = Sum[(-1)^j*Binomial[n-j, k]*Binomial[n-j+1, j]*CatalanNumber[n-k-j], {j, 0, n-k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 04 2021 *)

Sage

def A155839(n,k): return sum( (-1)^j*binomial(n-j,k)*binomial(n-j+1,j)*catalan_number(n-k-j) for j in (0..n-k))
flatten([[A155839(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 04 2021

Formula

T(n, k) = Sum_{j=k..n} (-1)^(n-j)*binomial(j+1, n-j)*binomial(j, k)*A000108(j-k).

Sum_{k=0..n} T(n, k) = A120010(n+1).

Equals A033184^{-1}*A124644.