A388052 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} binomial(n,j) * binomial(k*n+j+1,n).

1, 1, 3, 1, 5, 5, 1, 7, 25, 7, 1, 9, 61, 129, 9, 1, 11, 113, 575, 681, 11, 1, 13, 181, 1561, 5641, 3653, 13, 1, 15, 265, 3303, 22569, 56695, 19825, 15, 1, 17, 365, 6017, 63241, 335137, 579125, 108545, 17, 1, 19, 481, 9919, 143529, 1244979, 5064793, 5984767, 598417, 19

Offset

0,3

Links

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

Formula

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

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

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

A(n,k) = [x^n] (1+x)^(k*n+1) * (2+x)^n.

Example

Square array begins:
   1,    1,     1,      1,       1,       1, ...
   3,    5,     7,      9,      11,      13, ...
   5,   25,    61,    113,     181,     265, ...
   7,  129,   575,   1561,    3303,    6017, ...
   9,  681,  5641,  22569,   63241,  143529, ...
  11, 3653, 56695, 335137, 1244979, 3522221, ...

Prog

(PARI) a(n, k) = sum(j=0, n, binomial(n, j)*binomial(k*n+j+1, n));

Crossrefs

Columns k=0..4 give A005408, A002002(n+1), A108448(n+1), A388045, A388046.

Main diagonal gives A388053.

Cf. A341470, A387934.

Sequence in context: A111125 A209159 A182397 * A376102 A343510 A344725

Adjacent sequences: A388049 A388050 A388051 * A388053 A388054 A388055

Keyword

nonn,tabl

Author

Seiichi Manyama, Sep 14 2025

Status

approved