|
|
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.
|
|
13
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
LINKS
|
|
|
FORMULA
|
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, ...
|
|
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
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|