A152664 - OEIS

%I #8 Jul 21 2017 10:50:19

%S 1,1,1,4,2,12,8,4,72,36,12,360,216,108,36,2880,1440,576,144,20160,

%T 11520,5760,2304,576,201600,100800,43200,14400,2880,1814400,1008000,

%U 504000,216000,72000,14400,21772800,10886400,4838400,1814400,518400,86400

%N Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} for which k is the maximal number of initial even entries (0 <= k <= floor(n/2)).

%C Sum of entries in row n is n! (A000142).

%C Row n has 1 + floor(n/2) entries.

%C T(n,0) = A052558(n-1).

%C Sum_{k=0..ceiling(n/2)} k*T(n,k) = A152665(n).

%F T(2n+1,k) = n!(n+1)!binomial(2*n-k,n);

%F T(2n,k) = (n!)^2*binomial(2n-k-1,n-1).

%e T(3,0)=4 because we have 123, 132, 312 and 321.

%e T(4,2)=4 because we have 2413, 2431, 4213 and 4231.

%e Triangle starts:

%e     1;

%e     1,   1;

%e     4,   2;

%e    12,   8,   4;

%e    72,  36,  12;

%e   360, 216, 108,  36;

%p T := proc (n, k) if `mod`(n, 2) = 1 then factorial((1/2)*n-1/2)*factorial((1/2)*n+1/2)*binomial(n-k-1, (1/2)*n-1/2) else factorial((1/2)*n)^2*binomial(n-k-1, (1/2)*n-1) end if end proc: for n to 11 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

%Y Cf. A000142, A052558, A152662, A152665.

%K nonn,tabf

%O 1,4

%A _Emeric Deutsch_, Dec 13 2008