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%I #25 Feb 19 2024 01:50:46
%S 1,1,5,7,33,57,279,561,2895,6555,35685,89055,509985,1381905,8294895,
%T 24137505,151335135,468934515,3061162125,10033419375,68000295825,
%U 234484536825,1645756410375,5943863027025,43105900812975,162446292283275,1214871076343925,4761954230608575
%N a(n) = (n-1)!! - (n-2)!!.
%C For n >= 0 let A(n) be the product of the positive integers <= n that have the same parity as n minus the product of the positive integers <= n that have the opposite parity as n. Then a(n) = A(n-1) (for n >= 3). [_Peter Luschny_, Jul 06 2011]
%D S. P. Hurd and J. S. McCranie, Quantum factorials. Proceedings of the Twenty-fifth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1994). Congr. Numer. 104 (1994), 19-24.
%H T. D. Noe, <a href="/A007911/b007911.txt">Table of n, a(n) for n = 3..100</a>
%F (n-1)*a(n+2) = a(n+1) + n^2*a(n). - _Robert Israel_, Aug 08 2017
%p DDF := proc(n) local R, P, k; R := {$1..n}; P := select(k->k mod 2 = n mod 2, R); mul(k, k = P) - mul(k, k = R minus P) end: A007911 := n -> DDF(n-1); # _Peter Luschny_, Jul 06 2011
%p f:= gfun:-rectoproc({(-n+1)*a(2+n)+a(1+n)+n^2*a(n), a(2)=0,a(3)=1}, a(n), remember):
%p map(f, [$3..100]); # _Robert Israel_, Aug 08 2017
%t Table[(n - 1)!! - (n - 2)!!, {n, 3, 30}] (* _Vincenzo Librandi_, Aug 08 2017 *)
%o (Magma) DoubleFactorial:=func< n | &*[n..2 by -2] >; [DoubleFactorial((n-1))-DoubleFactorial(n-2): n in [3..30]]; // _Vincenzo Librandi_, Aug 08 2017
%Y Cf. A007912.
%K nonn,easy
%O 3,3
%A _J. H. Conway_
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