A358050 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(k*j,j) * binomial(k*(n-j),n-j).

1, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 6, 16, 4, 0, 1, 8, 39, 64, 5, 0, 1, 10, 72, 258, 256, 6, 0, 1, 12, 115, 664, 1719, 1024, 7, 0, 1, 14, 168, 1360, 6184, 11496, 4096, 8, 0, 1, 16, 231, 2424, 16265, 57888, 77052, 16384, 9, 0, 1, 18, 304, 3934, 35400, 195660, 543544, 517194, 65536, 10, 0

Offset

0,5

Links

Table of n, a(n) for n=0..65.

Formula

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

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

Example

Square array begins:
  1, 1,    1,     1,     1,      1, ...
  0, 2,    4,     6,     8,     10, ...
  0, 3,   16,    39,    72,    115, ...
  0, 4,   64,   258,   664,   1360, ...
  0, 5,  256,  1719,  6184,  16265, ...
  0, 6, 1024, 11496, 57888, 195660, ...

Prog

(PARI) T(n, k) = sum(j=0, n, binomial(k*j, j)*binomial(k*(n-j), n-j));
(PARI) T(n, k) = sum(j=0, n, (k-1)^(n-j)*binomial(k*n+1, j));
(PARI) T(n, k) = sum(j=0, n, k^(n-j)*binomial((k-1)*n+j, j));

Crossrefs

Column k=0-7 give: A000007, A001477(n+1), A000302, A006256, A078995, A079678, A079679, A079563.

Main diagonal gives A358145.

Cf. A358146.

Sequence in context: A357499 A368506 A342133 * A334781 A291656 A209063

Adjacent sequences: A358047 A358048 A358049 * A358051 A358052 A358053

Keyword

nonn,tabl

Author

Seiichi Manyama, Oct 31 2022

Status

approved