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Alternating sum of double factorials: n!! - (n-1)!! + (n-2)!! - ... 1!!.
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%I #18 Feb 02 2025 16:32:09

%S 1,1,2,6,9,39,66,318,627,3213,7182,38898,96237,548883,1478142,8843778,

%T 25615647,160178913,494550162,3221341038,10527969537,71221636863,

%U 245012506362,1716978047238,6188875533387,44822878860213,168635167816662,1259693955204138

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

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

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

%p A129831 := proc(n)

%p add( (-1)^i*doublefactorial(n-i),i=0..n-1) ;

%p end proc: # _R. J. Mathar_, Aug 22 2012

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n=0, 0, doublefactorial(n)-a(n-1)) end:

%p seq(a(n), n=1..30); # _Alois P. Heinz_, Feb 02 2025

%t Table[Sum[(-1)^i*(n-i)!!,{i,0,n-1}],{n,26}] (* _James C. McMahon_, Feb 02 2025 *)

%Y Cf. A005165 (similar for factorials), A006882.

%K nonn

%O 1,3

%A _Paolo P. Lava_ & _Giorgio Balzarotti_, May 21 2007