A174764 - OEIS

%I #30 Nov 04 2014 03:43:05

%S 0,1,3,18,70,330,1386,6328,28008,130140,603460,2895816,14024088,

%T 69786808,352043160,1817317440,9525774016,50958843408,276906491568,

%U 1532719442080,8615750596320,49260355141536,285887468809888

%N 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.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Probability_mass_function">Probability Mass Function</a>

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

%F a(n) = (n!/4*(n-4)!)*A000085(n-4) + (n!/2*(n-2)!)*A000085(n-2), n>3. - _Vale Murthy_, Nov 03 2014

%F a(n) = (n!/4*(n-4)!)*A000085(n-4) + A162970(n), n>3. - _Rajan Murthy_, Nov 03 2014

%F a(n) = (n!/2*(n-2)!)*A162970(n-2) + A162970(n), n>3. - _Rajan Murthy_, Nov 03 2014

%o (PARI) a(n) = sum(k=0, n\2 ,k^2*n!/((n-2*k)!*2^k*k!)); \\ _Michel Marcus_, Aug 10 2013

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

%K nonn

%O 1,3

%A _Rajan Murthy_, Nov 30 2010

%E More terms from _Michel Marcus_, Aug 10 2013