A387934 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,j).

1, 1, 2, 1, 3, 4, 1, 4, 13, 8, 1, 5, 26, 63, 16, 1, 6, 43, 190, 321, 32, 1, 7, 64, 416, 1462, 1683, 64, 1, 8, 89, 768, 4239, 11584, 8989, 128, 1, 9, 118, 1273, 9708, 44485, 93536, 48639, 256, 1, 10, 151, 1958, 19181, 126386, 475780, 765314, 265729, 512

Offset

0,3

Links

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

Formula

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

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

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

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

Example

Square array begins:
   1,    1,     1,      1,       1,       1,        1, ...
   2,    3,     4,      5,       6,       7,        8, ...
   4,   13,    26,     43,      64,      89,      118, ...
   8,   63,   190,    416,     768,    1273,     1958, ...
  16,  321,  1462,   4239,    9708,   19181,    34226, ...
  32, 1683, 11584,  44485,  126386,  297662,   616188, ...
  64, 8989, 93536, 475780, 1676956, 4707971, 11306572, ...

Prog

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

Crossrefs

Columns k=0..5 give A000079, A001850, A114496, A156886, A156887, A359646.

Main diagonal gives A306280.

Cf. A341470.

Sequence in context: A073135 A391485 A063804 * A213800 A388010 A224823

Adjacent sequences: A387931 A387932 A387933 * A387935 A387936 A387937

Keyword

nonn,tabl

Author

Seiichi Manyama, Sep 13 2025

Status

approved