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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).
2
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
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
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Oct 31 2022
STATUS
approved