A341470 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} binomial(k*n,n-j) * binomial(k*n+j,j).

1, 1, 1, 1, 3, 1, 1, 5, 13, 1, 1, 7, 41, 63, 1, 1, 9, 85, 377, 321, 1, 1, 11, 145, 1159, 3649, 1683, 1, 1, 13, 221, 2625, 16641, 36365, 8989, 1, 1, 15, 313, 4991, 50049, 246047, 369305, 48639, 1, 1, 17, 421, 8473, 118721, 982729, 3707509, 3800305, 265729, 1

Offset

0,5

Links

Table of n, a(n) for n=0..54.

Eric Weisstein's World of Mathematics, Delannoy Number.

Formula

T(n,k) = A008288(n,k*n).

T(n,k) = Sum_{j=0..n} 2^j * binomial(n,j) * binomial(k*n,j).

From Seiichi Manyama, Sep 13 2025: (Start)

T(n,k) = [x^n] (1-x)^n/(1-2*x)^(k*n+1).

T(n,k) = Sum_{j=0..n} 2^j * (-1)^(n-j) * binomial(n,j) * binomial(k*n+j,j).

T(n,k) = Sum_{j=0..n} binomial(n,j) * binomial(k*n+j,n). (End)

Example

Square array begins:
  1,    1,     1,      1,      1,       1, ...
  1,    3,     5,      7,      9,      11, ...
  1,   13,    41,     85,    145,     221, ...
  1,   63,   377,   1159,   2625,    4991, ...
  1,  321,  3649,  16641,  50049,  118721, ...
  1, 1683, 36365, 246047, 982729, 2908411, ...

Prog

(PARI) T(n, k) = sum(j=0, n, binomial(k*n, n-j)*binomial(k*n+j, j));
(PARI) T(n, k) = sum(j=0, n, 2^j*binomial(n, j)*binomial(k*n, j));

Crossrefs

Columns k=0..5 give A000012, A001850, A026000, A026001, A331329, A341491.

Rows n=0..2 give A000012, A005408, A102083.

Main diagonal gives A181675(n+1).

Cf. A008288, A387934.

Sequence in context: A294946 A083075 A335333 * A293796 A195892 A195522

Adjacent sequences: A341467 A341468 A341469 * A341471 A341472 A341473

Keyword

nonn,tabl

Author

Seiichi Manyama, Feb 13 2021

Status

approved