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%I #43 Sep 22 2023 02:13:08
%S 1,0,1,29,1721,163386,22831355,4420321081,1133879136649,
%T 372419001449076,152466248712342181,76134462292157828285,
%U 45552714996556390334921,32173493282909179882613934,26487410329744429030530295991,25143126122564855343240882599761,27260957330891104469298062949026065
%N Number of permutations of 3 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.
%H Seiichi Manyama, <a href="/A190826/b190826.txt">Table of n, a(n) for n = 0..223</a> (terms 0..101 from Andrew Woods)
%H H. Eriksson and A. Martin, <a href="http://arxiv.org/abs/1702.04177">Enumeration of Carlitz multipermutations</a>, arXiv:1702.04177 [math.CO], 2017.
%H R. J. Mathar, <a href="http://vixra.org/abs/1511.0015">A class of multinomial permutations avoiding object clusters</a>, vixra:1511.0015 (2015), sequence M_{c,3}/3!.
%F a(n) = A193624(n)/(6^n * n!), for n >= 1.
%F a(n) = A193638(n)/n!, for n >= 1.
%F a(n) = A192990(binomial(n+2,3)) / (6^n * n!), for n >= 1.
%F 2*a(n) -3*(3*n^2-3*n+4)*a(n-1) +2*(9*n^2-42*n+47)*a(n-2) +8*(3*n-7)*a(n-3) -8*a(n-4) = 0. - _R. J. Mathar_, May 23 2014
%F a(n) = (1/(6^n * n!)) * Sum_{j=0..2*n} Sum_{k=ceiling(j/2)..n} (n+j)! * binomial(2*k, j) * binomial(n, k) * (-3)^(n+k-j). - _Jean-François Alcover_, Jul 22 2017
%F a(n) ~ 3^(2*n + 1/2) * n^(2*n) / (2^n * exp(2*n + 2)). - _Vaclav Kotesovec_, Nov 24 2018
%e Some of the a(3) = 29 solutions for n=3: 123232131, 123121323, 123123213, 123212313, 123213123, 121323132, 123132312, 123123123, 123231213, 121323123, 121321323, 121312323, 121323231, 123231321, 121313232, 123132321, ...
%t a[n_]:= 1/(6^n*n!)*Sum[(n+j)! Sum[Binomial[n,k] Binomial[2k,j] (-3)^(n+k-j), {k, Ceiling[j/2], n}], {j,0,2n}]; Array[a, 16, 0] (* _Jean-François Alcover_, Jul 22 2017, after _Tani Akinari_'s code for A193638 *)
%o (Magma)
%o B:=Binomial;
%o f:= func< n,j | (&+[B(n,k)*B(2*k,j)*(-3)^(k-j): k in [Ceiling(j/2)..n]]) >;
%o A190826:= func< n | (-1/2)^n*(&+[Factorial(j)*B(n+j,j)*f(n,j): j in [0..2*n]]) >;
%o [A190826(n): n in [0..30]]; // _G. C. Greubel_, Sep 22 2023
%o (SageMath)
%o b=binomial;
%o def f(j,n): return sum(b(n,k)*b(2*k,j)*(-3)^(k-j) for k in range((j//2),n+1))
%o def A190826(n): return (-1/2)^n*sum(factorial(j)*b(n+j,j)*f(j,n) for j in range(2*n+1))
%o [A190826(n) for n in range(31)] # _G. C. Greubel_, Sep 22 2023
%Y Cf. A192990, A193624, A193638.
%Y Row n=3 of A322013.
%K nonn
%O 0,4
%A _R. H. Hardin_, May 21 2011
%E a(0)=1 prepended by _Alois P. Heinz_, Jul 22 2017
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