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

0,7

Links

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

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)

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 *)

Prog

(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
(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

Crossrefs

Cf. A056890, A058797.

Sequence in context: A029198 A029175 A186994 * A275761 A232396 A270096

Adjacent sequences: A056886 A056887 A056888 * A056890 A056891 A056892

Keyword

sign,frac,easy

Author

Aleksandar Petojevic, Sep 05 2000

Extensions

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

Status

approved