A209036 Number of permutations of the multiset {1,1,2,2,....,n,n} with exactly two consecutive equal terms.

1, 2, 36, 984, 43800, 2868480, 259554960, 31012490880, 4728875800320, 896042510496000, 206523228759724800, 56893926736333209600, 18461230471787348044800, 6968851610446509386803200

Offset

1,2

Comments

This is a particular case (p = 1) of the more general: a(p,n) = number of permutations of the multiset {1,1,2,2,....,n,n} with exactly p times two consecutive equal terms. The sequence a(0,p) is A114938.

Links

Table of n, a(n) for n=1..14.

Formula

a(1,1) = 1; a(p,n+1) = a[p, n + 1] = (2*n - p + 2)*a[p-1, n] + (2*n - p + 1)*(2*n - p)*a[p, n]/2 + p*a[p, n] + (p + 1)*(2*n - p)*a[p + 1, n + (p + 2)*(p + 1)*a[p + 2, n]/2.

Example

a(1,2) = 2, because 1221 and 2112 are the only permutations of {1,1,2,2} where exactly two consecutive terms are equal.

Prog

C-Language :
for (p = 0; p < 20; p++)
    a[p][0] = 0;
for (n = 0; n < 20; n++)
    a[0][n] = 0;
a[1][0] = 1;
for (n = 0; n < 18; n++)
   for (p = 0; p < 18; p++)
        a[p+1][n + 1] = (2*n - p + 2)*a[p][n] + (2*n - p + 1)*(2*n - p)*a[p+1][n]/2 + p*a[p+1][n] + (p + 1)*(2*n - p)*a[p + 2][n] + (p + 2)*(p + 1)*a[p + 3][n]/2 ;
   for(n = 0; n < 10; n++)
   {
    printf("%d, %ld     ", n, a[2][n]);
    if (n % 5 == 0)
     printf("\n\n");
   }

Crossrefs

Cf. A114938 (a(0,n)).

Sequence in context: A279575 A009539 A009554 * A375839 A305596 A210899

Adjacent sequences: A209033 A209034 A209035 * A209037 A209038 A209039

Keyword

nonn

Author

Philippe Gibone, Mar 04 2012

Status

approved