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A056890 Denominators of continued fraction for left factorial.

%I A056890 #8 Jul 02 2025 16:02:00
%S A056890 1,-1,0,-1,-2,1,1,0,1,-1,-4,3,14,-11,-63,52,353,-301,-2356,2055,18194,
%T A056890 -16139,-159335,143196,1559017,-1415821,-16846656,15430835,199185034,
%U A056890 -183754199,-2557127951,2373373752,35416852081,-33043478329,-526322279512,493278801183,8352696141782
%N A056890 Denominators of continued fraction for left factorial.
%H A056890 G. C. Greubel, <a href="/A056890/b056890.txt">Table of n, a(n) for n = 0..900</a>
%F A056890 a(0)=1; a(1)=-1; a(2*n)=n*a(2*n-1)+a(2*n-2); a(2*n+1)= - a(2*n)+a(2*n-1)
%p A056890 a:= proc(n) option remember;
%p A056890       if n<2 then (-1)^n
%p A056890     elif (n mod 2)=0 then (n/2)*a(n-1) +a(n-2)
%p A056890     else -a(n-1) +a(n-2)
%p A056890       fi; end:
%p A056890 seq(a(n), n=0..40); # _G. C. Greubel_, Dec 05 2019
%t A056890 a[n_]:= a[n]= If[n<2, (-1)^n, If[EvenQ[n], (n/2)*a[n-1] +a[n-2], -a[n-1] +a[n-2]]]; Table[a[n], {n,0,40}] (* _G. C. Greubel_, Dec 05 2019 *)
%o A056890 (PARI) a(n) = if(n<2, (-1)^n, if(Mod(n,2)==0, (n/2)*a(n-1) +a(n-2), -a(n-1) +a(n-2) )); \\ _G. C. Greubel_, Dec 05 2019
%o A056890 (Sage)
%o A056890 @CachedFunction
%o A056890 def a(n):
%o A056890     if (n<2): return (-1)^n
%o A056890     elif (mod(n,2) ==0): return (n/2)*a(n-1) +a(n-2)
%o A056890     else: return -a(n-1) +a(n-2)
%o A056890 [a(n) for n in (0..40)] # _G. C. Greubel_, Dec 05 2019
%o A056890 (GAP)
%o A056890 a:= function(n)
%o A056890     if n<2 then return (-1)^n;
%o A056890     elif (n mod 2)=0 then return (n/2)*a(n-1) +a(n-2);
%o A056890     else return -a(n-1) +a(n-2);
%o A056890     fi; end;
%o A056890 List([0..20], n-> a(n) ); # _G. C. Greubel_, Dec 05 2019
%Y A056890 Cf. A056889.
%K A056890 sign,frac,easy
%O A056890 0,5
%A A056890 _Aleksandar Petojevic_, Sep 05 2000
%E A056890 More terms from _James Sellers_, Sep 06 2000 and from Larry Reeves (larryr(AT)acm.org), Sep 07 2000