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

0,12

Links

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

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.

Example

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;

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A357712 A298159 A123735 * A229615 A359177 A135814

Adjacent sequences: A155836 A155837 A155838 * A155840 A155841 A155842

Keyword

easy,nonn,tabl

Author

Paul Barry, Jan 28 2009

Status

approved