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

1,2

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.

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.

Example

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, ...

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 *)

Prog

(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

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.

Sequence in context: A095832 A248162 A143381 * A277681 A140876 A243997

Adjacent sequences: A322090 A322091 A322092 * A322094 A322095 A322096

Keyword

nonn,tabl

Author

Seiichi Manyama, Nov 26 2018

Status

approved