OFFSET
0,2
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
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)
EXAMPLE
Triangle begins:
n
[0] 1;
[1] 2, 1;
[2] 6, 5, 2;
[3] 22, 21, 15, 5;
[4] 90, 89, 79, 49, 14;
...
MATHEMATICA
Flatten[Table[
Sum[Binomial[n + j, n] Binomial[n, j]/(j + 1), {j, k, n}], {n, 0,
10}, {k, 0, n}]]
PROG
(PARI) T(n, k)={sum(j=k, n, binomial(n+j, n)*binomial(n, j)/(j+1))} \\ Andrew Howroyd, Feb 09 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
David Callan, Feb 09 2022
STATUS
approved