A152665 -id:A152665 - cp's OEIS frontend

A152664 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)).

1, 1, 1, 4, 2, 12, 8, 4, 72, 36, 12, 360, 216, 108, 36, 2880, 1440, 576, 144, 20160, 11520, 5760, 2304, 576, 201600, 100800, 43200, 14400, 2880, 1814400, 1008000, 504000, 216000, 72000, 14400, 21772800, 10886400, 4838400, 1814400, 518400, 86400

Offset: 1

Emeric Deutsch, Dec 13 2008

Sum of entries in row n is n! (A000142).
Row n has 1 + floor(n/2) entries.
T(n,0) = A052558(n-1).
Sum_{k=0..ceiling(n/2)} k*T(n,k) = A152665(n).

			T(3,0)=4 because we have 123, 132, 312 and 321.
T(4,2)=4 because we have 2413, 2431, 4213 and 4231.
Triangle starts:
    1;
    1,   1;
    4,   2;
   12,   8,   4;
   72,  36,  12;
  360, 216, 108,  36;

Cf. A000142, A052558, A152662, A152665.

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

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

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

A152663 Number of leading odd entries in all permutations of {1,2,...,n} (see example).

Original entry on oeis.org

1, 1, 6, 16, 120, 540, 5040, 32256, 362880, 3024000, 39916800, 410572800, 6227020800, 76281004800, 1307674368000, 18598035456000, 355687428096000, 5762136335155200, 121645100408832000, 2211729098342400000, 51090942171709440000, 1030334000462807040000

Offset: 1

Emeric Deutsch, Dec 13 2008

nonn

a(n) = Sum_{k=0..ceiling(n/2)} k*A152662(n,k).

			a(3) = 6 because in the permutations 123, 132, 213, 231, 312, 321 we have 1+2+0+0+2+1 = 6 leading odd entries.

Cf. A152662, A152664, A152665.

ao := proc (n) options operator, arrow; factorial(2*n+1) end proc: ae := proc (n) options operator, arrow: n*factorial(2*n)/(n+1) end proc: a := proc (n) if `mod`(n, 2) = 1 then ao((1/2)*n-1/2) else ae((1/2)*n) end if end proc: seq(a(n), n = 1 .. 20);

a[n_] := If[OddQ[n], n!, n*n!/(n+2)];
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Apr 02 2024 *)

a(2n+1) = (2n+1)!;
a(2n) = n(2n)!/(n+1).
D-finite with recurrence 2*(n+2)*a(n) +3*(-n-1)*a(n-1) -2*n*(n-1)*(n+1)*a(n-2) +(n-2)*(n-1)^2*a(n-3)=0. - R. J. Mathar, Jul 26 2022

cp's OEIS Frontend

A152664 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)).

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