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

Author

Aleksandar Petojevic, Sep 05 2000

Keywords

• sign
• frac
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..900

Crossrefs

Cf. A056890, A058797.

Programs

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

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).

a(2*n) = (-1)^n * (A058797(n-2) + A058797(n-3)). (End)

Extensions

More terms from James Sellers, Sep 06 2000 and from Larry Reeves (larryr(AT)acm.org), Sep 07 2000