cp's OEIS Frontend

A056889 Numerators of continued fraction for left factorial.

%I A056889 #19 Jul 02 2025 16:02:00
%S A056889 0,1,1,0,1,-1,-2,1,2,-1,-3,2,9,-7,-40,33,224,-191,-1495,1304,11545,
%T A056889 -10241,-101106,90865,989274,-898409,-10690043,9791634,126392833,
%U A056889 -116601199,-1622625152,1506023953,22473758096,-20967734143,-333977722335,313009988192,5300202065121,-4987192076929
%N A056889 Numerators of continued fraction for left factorial.
%H A056889 G. C. Greubel, <a href="/A056889/b056889.txt">Table of n, a(n) for n = 0..900</a>
%F A056889 a(0) = 0; 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).
%F A056889 From _Mark van Hoeij_, Jul 15 2022: (Start)
%F A056889 a(2*n+1) = -(-1)^n * A058797(n-2).
%F A056889 a(2*n) = (-1)^n * (A058797(n-2) + A058797(n-3)). (End)
%p A056889 a:= proc(n) option remember;
%p A056889       if n<2 then n
%p A056889     elif (n mod 2)=0 then (n/2)*a(n-1) +a(n-2)
%p A056889     else -a(n-1) +a(n-2)
%p A056889       fi; end:
%p A056889 seq(a(n), n=0..40); # _G. C. Greubel_, Dec 05 2019
%t A056889 a[n_]:= a[n]= If[n<2, 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 A056889 (PARI) a(n) = if(n<2, 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 A056889 (Sage)
%o A056889 @CachedFunction
%o A056889 def a(n):
%o A056889     if (n<2): return n
%o A056889     elif (mod(n,2) ==0): return (n/2)*a(n-1) +a(n-2)
%o A056889     else: return -a(n-1) +a(n-2)
%o A056889 [a(n) for n in (0..40)] # _G. C. Greubel_, Dec 05 2019
%o A056889 (GAP)
%o A056889 a:= function(n)
%o A056889     if n<2 then return n;
%o A056889     elif (n mod 2)=0 then return (n/2)*a(n-1) +a(n-2);
%o A056889     else return -a(n-1) +a(n-2);
%o A056889     fi; end;
%o A056889 List([0..20], n-> a(n) ); # _G. C. Greubel_, Dec 05 2019
%Y A056889 Cf. A056890, A058797.
%K A056889 sign,frac,easy
%O A056889 0,7
%A A056889 _Aleksandar Petojevic_, Sep 05 2000
%E A056889 More terms from _James Sellers_, Sep 06 2000 and from Larry Reeves (larryr(AT)acm.org), Sep 07 2000