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

1,3

Links

Table of n, a(n) for n=1..23.

Wikipedia, Probability Mass Function

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

Prog

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

Crossrefs

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

Sequence in context: A373065 A374487 A098522 * A114633 A135070 A073961

Adjacent sequences: A174761 A174762 A174763 * A174765 A174766 A174767

Keyword

nonn

Author

Rajan Murthy, Nov 30 2010

Extensions

More terms from Michel Marcus, Aug 10 2013

Status

approved