A322093 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of permutations of n copies of 1..k with no element equal to another within a distance of 1.

1, 2, 0, 6, 2, 0, 24, 30, 2, 0, 120, 864, 174, 2, 0, 720, 39480, 41304, 1092, 2, 0, 5040, 2631600, 19606320, 2265024, 7188, 2, 0, 40320, 241133760, 16438575600, 11804626080, 134631576, 48852, 2, 0, 362880, 29083420800, 22278418248240, 131402141197200, 7946203275000, 8437796016, 339720, 2, 0

Offset: 1

Author

Seiichi Manyama, Nov 26 2018

Keywords

• nonn
• tabl

Examples

			Square array begins:
   1, 2,    6,        24,           120,                 720, ...
   0, 2,   30,       864,         39480,             2631600, ...
   0, 2,  174,     41304,      19606320,         16438575600, ...
   0, 2, 1092,   2265024,   11804626080,     131402141197200, ...
   0, 2, 7188, 134631576, 7946203275000, 1210527140790855600, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 1..52, flattened
• Mathematics.StackExchange, Find the number of k 1's, k 2's, ... , k n's - total kn cards, Apr 08 2012.

Crossrefs

Columns k=3 gives A110706.

Rows n=1..10 give A000142, A114938, A193638, A321633, A322126, A321382, A322095, A322096, A322145, A322146.

Main diagonal gives A321634.

Cf. A322013.

Programs

Mathematica

Table[Table[SeriesCoefficient[1/(1 - Sum[x[i]/(1 + x[i]), {i, 1, n}]), Sequence @@ Table[{x[i], 0, k}, {i, 1, n}]],{n, 1, 6}], {k, 1, 5}] (* Zlatko Damijanic, Nov 03 2024 *)

PARI

q(n,x) = sum(i=1, n, (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!)
T(n,k) = subst(serlaplace(q(n,x)^k), x, 1) \\ Andrew Howroyd, Feb 03 2024

Formula

A(n,k) = k! * A322013(n,k).

Let q_n(x) = Sum_{i=1..n} (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!.

A(n,k) = Integral_{0..infinity} (q_n(x))^k * exp(-x) dx.