A024419 - OEIS

%I #20 Oct 07 2016 00:17:16

%S 1,1,3,8,34,156,924,6144,48096,420480,4134240,44720640,530444160,

%T 6824805120,94787884800,1412038656000,22464536371200,380017225728000,

%U 6811416338227200,128936055177216000,2570286167543808000,53818546503794688000,1180914445357903872000

%N a(n) = n! (1/C(n,0) + 1/C(n,1) + ... + 1/C(n,[ n/2 ])).

%C Half-convolution of factorials (A000142) with itself. For the definition of the half-convolution of a sequence with itself see a comment to A201204. - _Vladimir Reshetnikov_, Oct 05 2016

%H Robert Israel, <a href="/A024419/b024419.txt">Table of n, a(n) for n = 0..449</a>

%F G.f.: (G(x)^2+H(x))/2 where G(x) = Sum_{k>=0} k!*x^k and H(x) = Sum_{k>=0} k!^2*x^(2*k). - _Vladeta Jovovic_, Sep 22 2007

%F a(n) = Sum_{k=0..floor(n/2)} k!*(n-k)!. - _Vladimir Reshetnikov_, Oct 05 2016

%e a(3)=3!*(1/1 + 1/3)=6*4/3=8.

%p a:=proc(n) options operator, arrow: factorial(n)*(sum(1/binomial(n, k), k= 0.. floor((1/2)*n))) end proc: seq(a(n), n=0..21); # _Emeric Deutsch_, Oct 11 2007

%t Table[Sum[k! (n - k)!, {k, 0, n/2}], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 05 2016 *)

%o (PARI) a(n) = sum(k=0, n\2, k!*(n-k)!); \\ _Michel Marcus_, Oct 05 2016

%Y Cf. A000142, A003149, A201204.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_

%E More terms from _Emeric Deutsch_, Oct 11 2007