A174764 Sum of the numerators for computing the second moment of the probability mass function (PMF) of the number of 2-cycles in the involutions on n elements (A000085) assuming the involutions are all equally likely.

0, 1, 3, 18, 70, 330, 1386, 6328, 28008, 130140, 603460, 2895816, 14024088, 69786808, 352043160, 1817317440, 9525774016, 50958843408, 276906491568, 1532719442080, 8615750596320, 49260355141536, 285887468809888

Offset: 1

Author

Rajan Murthy, Nov 30 2010

Keywords

• nonn

Links

• Wikipedia, Probability Mass Function

Crossrefs

First moment numerators are given by A162970. The denominator is given by A000085.

Programs

PARI

a(n) = sum(k=0, n\2 ,k^2*n!/((n-2*k)!*2^k*k!)); \\ Michel Marcus, Aug 10 2013

Formula

a(n) = Sum_{k=0..[ n/2 ]} k^2*n!/((n-2*k)!*2^k*k!).

a(n) = (n!/4*(n-4)!)*A000085(n-4) + (n!/2*(n-2)!)*A000085(n-2), n>3. - Vale Murthy, Nov 03 2014

a(n) = (n!/4*(n-4)!)*A000085(n-4) + A162970(n), n>3. - Rajan Murthy, Nov 03 2014

a(n) = (n!/2*(n-2)!)*A162970(n-2) + A162970(n), n>3. - Rajan Murthy, Nov 03 2014

Extensions

More terms from Michel Marcus, Aug 10 2013