A056889 Numerators of continued fraction for left factorial.
0, 1, 1, 0, 1, -1, -2, 1, 2, -1, -3, 2, 9, -7, -40, 33, 224, -191, -1495, 1304, 11545, -10241, -101106, 90865, 989274, -898409, -10690043, 9791634, 126392833, -116601199, -1622625152, 1506023953, 22473758096, -20967734143, -333977722335, 313009988192, 5300202065121, -4987192076929
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..900
Programs
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GAP
a:= function(n) if n<2 then return n; elif (n mod 2)=0 then return (n/2)*a(n-1) +a(n-2); else return -a(n-1) +a(n-2); fi; end; List([0..20], n-> a(n) ); # G. C. Greubel, Dec 05 2019 -
Maple
a:= proc(n) option remember; if n<2 then n elif (n mod 2)=0 then (n/2)*a(n-1) +a(n-2) else -a(n-1) +a(n-2) fi; end: seq(a(n), n=0..40); # G. C. Greubel, Dec 05 2019 -
Mathematica
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 *) -
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
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Sage
@CachedFunction def a(n): if (n<2): return n elif (mod(n,2) ==0): return (n/2)*a(n-1) +a(n-2) else: return -a(n-1) +a(n-2) [a(n) for n in (0..40)] # G. C. Greubel, Dec 05 2019
Formula
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).
From Mark van Hoeij, Jul 15 2022: (Start)
a(2*n+1) = -(-1)^n * A058797(n-2).
Extensions
More terms from James Sellers, Sep 06 2000 and from Larry Reeves (larryr(AT)acm.org), Sep 07 2000