A130318 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.

1, 1, 4, 128, 11520, 143360, 425425, 2064384, 619315200, 280284364800, 6801567252480, 27512370460575, 178541140377600, 152355106455552000, 167834385271436083200, 6074006324109115392000, 29734853645550994565625, 231916605102348042240000, 392866729043377583554560000

Offset

0,3

Comments

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

Links

Jayanta Basu, Table of n, a(n) for n = 0..100

Formula

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]

Example

6 --> 6!! = 48; 6 + 4 + 2 = 12; 48/12 = 4.
17 --> 17!! = 34459425; 17 + 15 + 13 + 11 + 9 + 7 + 5 + 3 + 1 = 81; 34459425/81 = 425425.

Maple

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);
# second Maple program:
f:= n-> `if`(irem(doublefactorial(n), floor((n+1)^2/4), 'r')=0, r, [][]):
map(f, [$1..50])[];  # Alois P. Heinz, Mar 16 2024

Mathematica

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 *)

Crossrefs

Cf. A108552, A000290, A005408, A130319.

Cf. A002620, A006882.

Sequence in context: A267796 A013823 A321233 * A000318 A229385 A141367

Adjacent sequences: A130315 A130316 A130317 * A130319 A130320 A130321

Keyword

easy,nonn

Author

Paolo P. Lava and Giorgio Balzarotti, May 23 2007

Status

approved