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%I #6 Jul 23 2019 02:03:24
%S 1,1,0,1,1,0,2,3,0,9,8,3,44,45,12,1,265,264,90,8,1854,1855,660,90,2,
%T 14833,14832,5565,880,45,133496,133497,51912,9275,660,9,1334961,
%U 1334960,533988,103824,9275,264,14684570,14684571,6007320,1245972,129780,5565
%N Triangle read by rows: T(n,k) is the number of permutations of [n] starting with 1, having no 3-sequences and having k successions (0 <= k <= floor(n/2)); a succession of a permutation p is a position i such that p(i +1) - p(i) = 1.
%C Row n has 1+floor(n/2) entries.
%C Sum of entries in row n is A165961(n).
%C T(n,0) = d(n-1).
%C Sum_{k>=0} k*T(n,k) = A180187(n).
%C From _Emeric Deutsch_, Sep 07 2010: (Start)
%C T(n,k) is also the number of permutations of [n-1] with k fixed points, no two of them adjacent. Example: T(5,2)=3 because we have 1432, 1324, and 3214.
%C (End)
%F T(n,k) = binomial(n-k,k)*d(n-k-1), where d(j) = A000166(j) are the derangement numbers.
%e T(5,2)=3 because we have 12453, 12534, and 14523.
%e Triangle starts:
%e 1;
%e 1;
%e 0, 1;
%e 1, 0;
%e 2, 3, 0;
%e 9, 8, 3;
%e 44, 45, 12, 1;
%e 265, 264, 90, 8;
%p d[0] := 1: for n to 51 do d[n] := n*d[n-1]+(-1)^n end do: a := proc (n, k) if n = 0 and k = 0 then 1 elif k <= (1/2)*n then binomial(n-k, k)*d[n-1-k] else 0 end if end proc: for n from 0 to 12 do seq(a(n, k), k = 0 .. (1/2)*n) end do; # yields sequence in triangular form
%Y Cf. A000166, A165961, A180187.
%K nonn,tabf
%O 0,7
%A _Emeric Deutsch_, Sep 06 2010
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