A351385 Triangle read by rows: T(n,k) = Sum_{j=k..n} binomial(n + j, n)*binomial(n, j)/(j + 1).

1, 2, 1, 6, 5, 2, 22, 21, 15, 5, 90, 89, 79, 49, 14, 394, 393, 378, 308, 168, 42, 1806, 1805, 1784, 1644, 1224, 594, 132, 8558, 8557, 8529, 8277, 7227, 4917, 2145, 429, 41586, 41585, 41549, 41129, 38819, 31889, 19877, 7865, 1430, 206098, 206097, 206052, 205392, 200772, 182754, 140712, 80652, 29172, 4862

Offset: 0

Author

David Callan, Feb 09 2022

Keywords

• nonn
• tabl

Comments

T(n,k) is the number of central Delannoy paths of steps E = (1,0), N = (0,1), D = (1,1) from the origin to (n,n) with k E steps above the diagonal line y=x. For example, T(3,1) = 5 counts ENNE, NEEN, NED, NDE, DNE. That the titular sum counts these paths is a consequence of the following equidistribution result: among the central Delannoy n-paths with j E steps, the statistic "number of E steps above y=x" is uniformly distributed over {0,1,...,j}. So, for k <= j <= n, there are binomial(n + j, n) binomial(n, j)/(j + 1) central Delannoy n-paths with j E steps, k of which are above y = x.

Examples

			Triangle begins:
   n
  [0]  1;
  [1]  2,  1;
  [2]  6,  5,  2;
  [3] 22, 21, 15,  5;
  [4] 90, 89, 79, 49, 14;
      ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Crossrefs

Columns k=0..1 give: A006318, A035011.

Main diagonal gives A000108.

Row sums give A001850.

Cf. A001003, A002695, A080243, A088617 gives summands in title.

Programs

Mathematica

Flatten[Table[
  Sum[Binomial[n + j, n] Binomial[n, j]/(j + 1), {j, k, n}], {n, 0,
   10}, {k, 0, n}]]

PARI

T(n,k)={sum(j=k, n, binomial(n+j, n)*binomial(n,j)/(j+1))} \\ Andrew Howroyd, Feb 09 2022

Formula

G.f.: 2/(sqrt(1 - 6*x + x^2) + sqrt(1 - 2*x + x^2 - 4*x*y)).

From Alois P. Heinz, Feb 09 2022: (Start)

Sum_{k=0..n} k * T(n,k) = A002695(n).

Sum_{k=0..n} (-1)^k * T(n,k) = A001003(n).

Sum_{k=0..n} (-1)^k * T(n,n-k) = A080243(n). (End)