Aleksandar Petojevic - cp's OEIS frontend

A056158 Equivalent of the Kurepa hypothesis for left factorial.

-4, -2, -4, 2, -20, 86, -532, 3706, -29668, 266990, -2669924, 29369138, -352429684, 4581585862, -64142202100, 962133031466, -15394128503492, 261700184559326, -4710603322067908, 89501463119290210, -1790029262385804244

Offset: 3

Aleksandar Petojevic, Jul 31 2000

For a prime p > 2 we have !p == -a(p) mod p, where the left factorial !n = Sum_{k=0..n-1} k! (A003422).

James Spahlinger, Table of n, a(n) for n = 3..100
Romeo Mestrovic, Variations of Kurepa's left factorial hypothesis, arXiv:1312.7037 [math.NT], 2013.
Romeo Mestrovic, The Kurepa-Vandermonde matrices arising from Kurepa's left factorial hypothesis, Filomat 29:10 (2015), 2207-2215; DOI 10.2298/FIL1510207M.

[n eq 3 select -4 else -(n-3)*Self(n-3)-2*(n-1): n in [3..30]]; // Vincenzo Librandi, Feb 22 2016

a[3] = -4; a[n_]:= -(n-3)*a[n-1] - 2*(n-1); Array[a, 30, 3] (* James Spahlinger, Feb 20 2016 *)
Drop[CoefficientList[Series[2*x^2*(Exp[1/x -1]*ExpIntegralEi[(x-1)/x] + x/(x-1)), {x,0,15}, Assumptions -> x > 0], x],3] (* G. C. Greubel, Mar 29 2019 *)

m=30; v=concat([-4], vector(m-1)); for(n=2, m, v[n]=-(n-1)*v[n-1] -2*(n+1)); v \\ G. C. Greubel, Mar 29 2019

@CachedFunction
def Self(n):
   if n == 3 : return -4
   return -(n-3)*Self(n-1) - 2*(n-1)
[Self(n) for n in (3..30)] # G. C. Greubel, Mar 29 2019

a(3) = -4, a(n) = -(n-3)*a(n-1) - 2*(n-1).
a(n) = 2*(-1)^(n-1)*(n-3)!*Sum_{k=0..n-3} frac((k+2)*(-1)^(k+1))*k!.
Conjecture: a(n) + (n-5)*a(n-1) + (-2*n+9)*a(n-2) + (n-5)*a(n-3) = 0. - R. J. Mathar, Jan 31 2014
a(n) ~ (-1)^n * 2 * exp(-1) * (n-3)!. - Vaclav Kotesovec, Jan 05 2019
G.f.: 2*x^2*(exp(-1+1/x) * Exponential-Integral((x-1)/x) + x/(x-1)). - G. C. Greubel, Mar 29 2019

More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2000

A056953 Denominators of continued fraction for alternating factorial.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 34, 73, 209, 501, 1546, 4051, 13327, 37633, 130922, 394353, 1441729, 4596553, 17572114, 58941091, 234662231, 824073141, 3405357682, 12470162233, 53334454417, 202976401213, 896324308634, 3535017524403, 16083557845279, 65573803186921

Offset: 0

Aleksandar Petojevic, Sep 05 2000

Starting (1, 2, 3, ...) with offset 0 = eigensequence of an infinite lower triangular matrix with 1's in the main diagonal and the natural numbers repeated in the subdiagonal. - Gary W. Adamson, Feb 14 2011

a(n) is the number of involutions of [n] such that every 2-cycle contains one odd and one even element; a(4) = 7: 1234, 1243, 1324, 2134, 2143, 4231, 4321. - Alois P. Heinz, Feb 14 2013

Alois P. Heinz, Table of n, a(n) for n = 0..400
Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, Ramified inverse and planar monoids, arXiv:2210.17461 [math.RT], 2022.
Index entries for sequences related to Laguerre polynomials

Bisections are A000262 and A002720.

Cf. A124428, diagonals of A088699.

[(&+[Factorial(k)*Binomial(Floor(n/2),k)*Binomial(Floor((n+1)/2),k): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, May 16 2018

a:= proc(n) option remember; `if`(n<4, [1, 1, 2, 3][n+1],
      ((4*n-2)*a(n-2) +2*a(n-3) -(n-2)*(n-3)*a(n-4)) /4)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 14 2013

Table[Sum[k!*Binomial[Floor[n/2], k]*Binomial[Floor[(n+1)/2], k] , {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, May 16 2018 *)

a(n)=sum(k=0,n\2,k!*binomial(n\2,k)*binomial((n+1)\2,k)) \\ Paul D. Hanna, Oct 31 2006

a(0)=1; a(1)=1; a(n) = a(n-1) + n*a(n-2)/2.
a(n) = Sum_{k=0..[n/2]} k!*C([n/2],k)*C([(n+1)/2],k). - Paul D. Hanna, Oct 31 2006
a(n) ~ n^(n/2 + 1/4) / (2^(n/2 + 3/4) * exp(n/2 - sqrt(2*n) + 1/2)) * (1 + (25 + 6*(-1)^n)/(24*sqrt(2*n)) + (397 + 156*(-1)^n)/(2304*n)). - Vaclav Kotesovec, Feb 22 2019

A056889 Numerators of continued fraction for left factorial.

Original entry on oeis.org

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

Aleksandar Petojevic, Sep 05 2000

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

Cf. A056890, A058797.

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

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

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 *)

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

@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

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)

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

A056890 Denominators of continued fraction for left factorial.

Original entry on oeis.org

1, -1, 0, -1, -2, 1, 1, 0, 1, -1, -4, 3, 14, -11, -63, 52, 353, -301, -2356, 2055, 18194, -16139, -159335, 143196, 1559017, -1415821, -16846656, 15430835, 199185034, -183754199, -2557127951, 2373373752, 35416852081, -33043478329, -526322279512, 493278801183, 8352696141782

Offset: 0

Aleksandar Petojevic, Sep 05 2000

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

Cf. A056889.

a:= function(n)
    if n<2 then return (-1)^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

a:= proc(n) option remember;
      if n<2 then (-1)^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

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 *)

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

@CachedFunction
def a(n):
    if (n<2): return (-1)^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

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)

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

A056919 Numerators of continued fraction for left factorial.

Original entry on oeis.org

0, 1, 1, 0, -2, -2, 4, 10, -6, -46, -16, 214, 310, -974, -3144, 3674, 28826, -566, -260000, -254906, 2345094, 4894154, -20901880, -74737574, 176084986, 1072935874, -1216168944, -15164335306, 1862029910, 214162724194, 186232275544, -3026208587366, -6005924996070, 42413412401786

Offset: 0

Aleksandar Petojevic, Sep 05 2000

a(0)=0; a(1)=1; a(n) = a(n-1) - floor(n/2)*a(n-2).

More terms from James Sellers, Sep 06 2000

A056920 Denominators of continued fraction for left factorial.

Original entry on oeis.org

1, 1, 0, -1, -1, 1, 4, 1, -15, -19, 56, 151, -185, -1091, 204, 7841, 6209, -56519, -112400, 396271, 1520271, -2442439, -19165420, 7701409, 237686449, 145269541, -2944654296, -4833158329, 36392001815, 104056218421, -441823808804, -2002667085119, 5066513855745, 37109187217649

Offset: 0

Aleksandar Petojevic, Sep 05 2000

G. C. Greubel, Table of n, a(n) for n = 0..895

a:= function(n)
    if n<2 then return 1;
    else return a(n-1) - Int(n/2)*a(n-2);
    fi; end;
List([0..40], n-> a(n) ); # G. C. Greubel, Dec 05 2019

function a(n)
  if n lt 2 then return 1;
  else return a(n-1) - Floor(n/2)*a(n-2);
  end if; return a; end function;
[a(n): n in [0..40]]; // G. C. Greubel, Dec 05 2019

a:= proc(n) option remember;
   if n<2 then 1
   else a(n-1) - floor(n/2)*a(n-2)
   fi; end:
seq(a(n), n=0..40); # G. C. Greubel, Dec 05 2019

a[n_]:= a[n]= If[n<2, 1, a[n-1] -Floor[n/2]*a[n-2]]; Table[a[n], {n,0,40}] (* G. C. Greubel, Dec 05 2019 *)

a(n) = if(n<2, n, a(n-1) - (n\2)*a(n-2) ); \\ G. C. Greubel, Dec 05 2019

@CachedFunction
def a(n):
    if (n<2): return 1
    else: return a(n-1) - floor(n/2)*a(n-2)
[a(n) for n in (0..40)] # G. C. Greubel, Dec 05 2019

a(0)=1, a(1)=1, a(n) = a(n-1) - floor(n/2)*a(n-2).

More terms from James Sellers, Sep 06 2000

A056921 a(0) = 0, a(1) = 1, a(2n) = na(2n-1) + a(2n-2), a(2n+1) = a(2n) + a(2*n-1).

Original entry on oeis.org

0, 1, 1, 2, 5, 7, 26, 33, 158, 191, 1113, 1304, 8937, 10241, 80624, 90865, 807544, 898409, 8893225, 9791634, 106809565, 116601199, 1389422754, 1506023953, 19461710190, 20967734143, 292042254049, 313009988192, 4674182088737

Offset: 0

Aleksandar Petojevic, Sep 05 2000

Seiichi Manyama, Table of n, a(n) for n = 0..897

Cf. A058797.

Numerators of continued fraction for alternating factorial.

More terms from James Sellers, Sep 07 2000

A056922 Denominators of continued fraction for alternating factorial.

Original entry on oeis.org

1, 1, 2, 3, 8, 11, 41, 52, 249, 301, 1754, 2055, 14084, 16139, 127057, 143196, 1272625, 1415821, 14015014, 15430835, 168323364, 183754199, 2189619553, 2373373752, 30670104577, 33043478329, 460235322854, 493278801183

Offset: 0

Aleksandar Petojevic, Sep 05 2000

Seiichi Manyama, Table of n, a(n) for n = 0..897

Cf. A058799, A056952 (numerators).

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).

More terms from James Sellers, Sep 07 2000

A056952 Numerators of continued fraction for alternating factorial.

Original entry on oeis.org

0, 1, 1, 2, 4, 8, 20, 44, 124, 300, 920, 2420, 7940, 22460, 78040, 235260, 859580, 2741660, 10477880, 35152820, 139931620, 491459820, 2030707640, 7436765660, 31805257340, 121046445260, 534514790680, 2108118579060, 9591325648580

Offset: 0

Aleksandar Petojevic, Sep 05 2000

Cf. A002793, A056922 (denominators).

a(0)=0; a(1)=1; a(n) = a(n-1) + (n/2)*a(n-2).

a(2n) = A002793(n); a(2n-1) = A002793(n) - n * A002793(n-1). - Max Alekseyev, Jul 07 2010

More terms from James Sellers, Sep 07 2000

A052169 Equivalent of the Kurepa hypothesis for left factorial.

Original entry on oeis.org

1, 2, 5, 19, 91, 531, 3641, 28673, 254871, 2523223, 27526069, 328018989, 4239014627, 59043418019, 881715042417, 14052333488521, 238063061452591, 4271909380510383, 80941440893880941, 1614781745832924773, 33833522293642233339, 742799603083145395579

Offset: 2

Aleksandar Petojevic, Jan 26 2000

Alois P. Heinz, Table of n, a(n) for n = 2..450
Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019. See Table 1, p. 6.
Sergi Elizalde, Bijections for restricted inversion sequences and permutations with fixed points, arXiv:2006.13842 [math.CO], 2020.
T. Kotek and J. A. Makowsky, Recurrence Relations for Graph Polynomials on Bi-iterative Families of Graphs, arXiv preprint arXiv:1309.4020 [math.CO], 2013.
Romeo Mestrovic, Variations of Kurepa's left factorial hypothesis, arXiv preprint arXiv:1312.7037 [math.NT], 2013.
Romeo Mestrovic, The Kurepa-Vandermonde matrices arising from Kurepa's left factorial hypothesis, Filomat 29:10 (2015), 2207-2215; DOI 10.2298/FIL1510207M.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.

Pairwise sums of A002467.

a[2] := 1: a[3] := 2: for n from 4 to 21 do a[n] := (n-2)*a[n-1]+(n-3)*a[n-2] end do: seq(a[n], n = 2 .. 21); # Emeric Deutsch, Jun 15 2009
# second Maple program:
a:= proc(n) option remember; `if`(n<4, n-1,
      (n-2)*a(n-1)+(n-3)*a(n-2))
    end:
seq(a(n), n=2..25);  # Alois P. Heinz, Aug 30 2016

Numerator[k=1; NestList[1+1/(k++ #1)&,1,12]] (* Wouter Meeussen, Mar 24 2007 *)
a[n_] := (n! - Subfactorial[n])/(n-1); Table[a[n], {n, 2, 23}] (* Jean-François Alcover, Jul 21 2017, after Emeric Deutsch's comment *)

from sage.combinat.sloane_functions import ExtremesOfPermanentsSequence2 ; e = ExtremesOfPermanentsSequence2() ; it = e.gen(1,2,1) ; [next(it) for i in range(20)] #(5 rows) # Zerinvary Lajos, May 15 2009

a(2) = 1, a(3) = 2, a(n) = (n-2)*a(n-1) + (n-3)*a(n-2).
a(n) = A002467(n)/(n-1) (A002467(n) = number of non-derangements of {1,2,...,n}). - Emeric Deutsch, Jun 15 2009
a(n) = 2*floor((n+1)!*(n+3)/e+1/2) - 3*(floor(((n+1)!+1)/e)+ floor(((n+2)!+1)/e)) +(n+1)!+(n+2)!, n>1, with offset 0..a(0)= 1. - Gary Detlefs, Apr 18 2010
a(n) = 1/(n+1)*((n+2)!-floor(((n+2)!+1)/e)), with offset 0 a(n) = 1/(n-1)*(n! - floor((n!+1)/e)). - Gary Detlefs, Jul 11 2010
From Benedict W. J. Irwin, Jun 02 2016: (Start)
Let y(-1)=1, y(0)=1, and y(n) = (Sum_{k=0..n-1} y(k)+y(k-1))/n,
a(n) = (n-2)!*y(n-2).
(End)
a(n) = (Gamma(n+1,0)-exp(-1)*Gamma(n+1,-1))/(n-1). - Martin Clever, Mar 25 2023

More terms from James Sellers, Jan 31 2000

cp's OEIS Frontend

User: Aleksandar Petojevic

A056158 Equivalent of the Kurepa hypothesis for left factorial.

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A056953 Denominators of continued fraction for alternating factorial.

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A056889 Numerators of continued fraction for left factorial.

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A056890 Denominators of continued fraction for left factorial.

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A056919 Numerators of continued fraction for left factorial.

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A056920 Denominators of continued fraction for left factorial.

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A056921 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).

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A056921 a(0) = 0, a(1) = 1, a(2n) = na(2n-1) + a(2n-2), a(2n+1) = a(2n) + a(2*n-1).