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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.

%I A322093 #43 Nov 13 2024 08:30:16
%S A322093 1,2,0,6,2,0,24,30,2,0,120,864,174,2,0,720,39480,41304,1092,2,0,5040,
%T A322093 2631600,19606320,2265024,7188,2,0,40320,241133760,16438575600,
%U A322093 11804626080,134631576,48852,2,0,362880,29083420800,22278418248240,131402141197200,7946203275000,8437796016,339720,2,0
%N 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.
%H A322093 Seiichi Manyama, <a href="/A322093/b322093.txt">Antidiagonals n = 1..52, flattened</a>
%H A322093 Mathematics.StackExchange, <a href="https://math.stackexchange.com/questions/129451/find-the-number-of-arrangements-of-k-mbox-1s-k-mbox-2s-cdots">Find the number of k 1's, k 2's, ... , k n's - total kn cards</a>, Apr 08 2012.
%F A322093 A(n,k) = k! * A322013(n,k).
%F A322093 Let q_n(x) = Sum_{i=1..n} (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!.
%F A322093 A(n,k) = Integral_{0..infinity} (q_n(x))^k * exp(-x) dx.
%e A322093 Square array begins:
%e A322093    1, 2,    6,        24,           120,                 720, ...
%e A322093    0, 2,   30,       864,         39480,             2631600, ...
%e A322093    0, 2,  174,     41304,      19606320,         16438575600, ...
%e A322093    0, 2, 1092,   2265024,   11804626080,     131402141197200, ...
%e A322093    0, 2, 7188, 134631576, 7946203275000, 1210527140790855600, ...
%t A322093 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 *)
%o A322093 (PARI)
%o A322093 q(n,x) = sum(i=1, n, (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!)
%o A322093 T(n,k) = subst(serlaplace(q(n,x)^k), x, 1) \\ _Andrew Howroyd_, Feb 03 2024
%Y A322093 Columns k=3 gives A110706.
%Y A322093 Rows n=1..10 give A000142, A114938, A193638, A321633, A322126, A321382, A322095, A322096, A322145, A322146.
%Y A322093 Main diagonal gives A321634.
%Y A322093 Cf. A322013.
%K A322093 nonn,tabl
%O A322093 1,2
%A A322093 _Seiichi Manyama_, Nov 26 2018