A129831 Alternating sum of double factorials: n!! - (n-1)!! + (n-2)!! - ... 1!!.

1, 1, 2, 6, 9, 39, 66, 318, 627, 3213, 7182, 38898, 96237, 548883, 1478142, 8843778, 25615647, 160178913, 494550162, 3221341038, 10527969537, 71221636863, 245012506362, 1716978047238, 6188875533387, 44822878860213, 168635167816662, 1259693955204138

Offset: 1

Author

Paolo P. Lava & Giorgio Balzarotti, May 21 2007

Keywords

• nonn

Examples

			a(5) = 5!! - 4!! + 3!! - 2!! + 1!! = 15 - 8 + 3 - 2 + 1 = 9.
		

Crossrefs

Cf. A005165 (similar for factorials), A006882.

Programs

Maple

A129831 := proc(n)
        add( (-1)^i*doublefactorial(n-i),i=0..n-1) ;
end proc: # R. J. Mathar, Aug 22 2012
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 0, doublefactorial(n)-a(n-1)) end:
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 02 2025

Mathematica

Table[Sum[(-1)^i*(n-i)!!,{i,0,n-1}],{n,26}] (* James C. McMahon, Feb 02 2025 *)

Formula

a(n) = n!! - (n-1)!! + (n-2)!! - ... 1!! = A006882(n) - a(n-1).