A130318 - OEIS

%I #21 Mar 16 2024 21:01:42

%S 1,1,4,128,11520,143360,425425,2064384,619315200,280284364800,

%T 6801567252480,27512370460575,178541140377600,152355106455552000,

%U 167834385271436083200,6074006324109115392000,29734853645550994565625,231916605102348042240000,392866729043377583554560000

%N Integer values of k!!/S(k), where S(k) is the sum of all odd numbers less than or equal to k, if k is odd, or the sum of all even numbers less than or equal to k, if k is even.

%C For n >= 8, a(n) ends with 0 or 5.

%H Jayanta Basu, <a href="/A130318/b130318.txt">Table of n, a(n) for n = 0..100</a>

%F Integers of the form k!!/((k+1)/2)^2, for k odd and k!!/(k*(k+2)/4) for k even. [corrected by _Jon E. Schoenfield_, Mar 16 2024]

%e 6 --> 6!! = 48; 6 + 4 + 2 = 12; 48/12 = 4.

%e 17 --> 17!! = 34459425; 17 + 15 + 13 + 11 + 9 + 7 + 5 + 3 + 1 = 81; 34459425/81 = 425425.

%p P:=proc(n) local a,i,j,k,w; for i from 1 by 1 to n do k:=i; w:=i-2; while w>0 do k:=k*w; w:=w-2; od; j:=i; w:=i-2; while w>0 do j:=j+w; w:=w-2; od; a:=k/j; if trunc(a)=a then print(a) fi; od; end: P(100);

%p # second Maple program:

%p f:= n-> `if`(irem(doublefactorial(n), floor((n+1)^2/4), 'r')=0, r, [][]):

%p map(f, [$1..50])[];  # _Alois P. Heinz_, Mar 16 2024

%t Select[Table[Times @@ (t = If[OddQ[n], Range[1, n, 2], Range[2, n, 2]])/Plus @@ t, {n, 41}], IntegerQ] (* _Jayanta Basu_, Aug 12 2013 *)

%Y Cf. A108552, A000290, A005408, A130319.

%Y Cf. A002620, A006882.

%K easy,nonn

%O 0,3

%A _Paolo P. Lava_ and _Giorgio Balzarotti_, May 23 2007