cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-18 of 18 results.

A129135 Number of permutations of [n] with exactly 5 fixed points.

Original entry on oeis.org

1, 0, 21, 112, 1134, 11088, 122430, 1468368, 19090071, 267258992, 4008887883, 64142201760, 1090417436108, 19627513841376, 372922762997772, 7458455259939936, 156627560458759005, 3445806330092671776, 79253545592131484497, 1902085094211155585424
Offset: 5

Views

Author

Zerinvary Lajos, May 25 2007

Keywords

Links

Crossrefs

Cf. A008290, A008291, A170942.

Programs

  • Maple
    a:=n->sum((n-1)!*sum((-1)^k/(k-4)!, j=0..n-1), k=4..n-1)/5!: seq(a(n), n=5..24);
x:='x'; G(x):=exp(-x)/(1-x)*(x^5/5!): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=5..24); # Zerinvary Lajos, Apr 03 2009
a:= n-> simplify(pochhammer(6, n-5)*GAMMA(n-4, -1)*exp(-1)/GAMMA(n-4)):
seq(a(n), n = 5 .. 24); # Miles Wilson, Aug 04 2024
  • Mathematica
    With[{nn=30},Drop[CoefficientList[Series[Exp[-x]/(1-x) x^5/5!,{x,0,nn}],x]Range[0,nn]!,5]] (* Harvey P. Dale, Jan 22 2013 *)
  • PARI
    my(x='x+O('x^66)); Vec(serlaplace(exp(-x)/(1-x)*(x^5/5!))) \\ Joerg Arndt, Feb 17 2014
  • Python
    from sympy import binomial
A129135_list, m, x = [], 1, 0
for n in range(5,21):
    x, m = x*n + m*binomial(n,5), -m
    A129135_list.append(x) # Chai Wah Wu, Nov 01 2014

Formula

a(n) = A008290(n,5).
E.g.f.: exp(-x)/(1-x)*(x^5/5!). - Zerinvary Lajos, Apr 03 2009
a(n) = n*a(n-1) - (-1^n)*binomial(n,5) with a(n) = 0 for n = 0,1,2,3,4. - Chai Wah Wu, Nov 01 2014
D-finite with recurrence (-n+5)*a(n) +n*(n-6)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 02 2015
O.g.f.: (1/5!)*Sum_{k>=5} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 15 2017

Extensions

Offset corrected by Susanne Wienand, Feb 17 2014

A129218 Permutations with exactly 10 fixed points.

Original entry on oeis.org

1, 0, 66, 572, 9009, 132132, 2122120, 36056592, 649062414, 12332093488, 246642054516, 5179482792120, 113948622073286, 2620818306541512, 62899639358957544, 1572490983970669840, 40884765583242727575
Offset: 10

Views

Author

Zerinvary Lajos, May 25 2007

Keywords

Links

Crossrefs

Cf. A008290, A008291, A170942.

Programs

  • Maple
    a:=n->sum(n!*sum((-1)^k/(k-9)!, j=0..n), k=9..n): seq(-a(n)/10!, n=9..27);
restart: G(x):=exp(-x)/(1-x)*(x^10/10!): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=10..26); # Zerinvary Lajos, Apr 03 2009
  • Mathematica
    With[{nn=40}, Drop[CoefficientList[Series[Exp[-x]/(1 - x) x^10/10!, {x, 0, nn}], x]Range[0, nn]!, 10]] (* Vincenzo Librandi, Feb 19 2014 *)
  • PARI
    x='x+O('x^66); Vec( serlaplace(exp(-x)/(1-x)*(x^9/9!)) ) \\ Joerg Arndt, Feb 19 2014

Formula

a(n) = A008290(n,10).
E.g.f.: exp(-x)/(1-x)*(x^10/10!). [Zerinvary Lajos, Apr 03 2009]
O.g.f.: (1/10!)*Sum_{k>=10} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 15 2017

Extensions

Changed offset from 0 to 10 by Vincenzo Librandi, Feb 19 2014
Edited by Joerg Arndt, Feb 19 2014

A129136 Permutations with exactly 6 fixed points.

Original entry on oeis.org

1, 0, 28, 168, 1890, 20328, 244860, 3181464, 44543499, 668147480, 10690367688, 181736238320, 3271252308324, 62153793831024, 1243075876659240, 26104593409789776, 574301055015449685, 13208924265355241808
Offset: 6

Views

Author

Zerinvary Lajos, May 25 2007

Keywords

Links

Crossrefs

Cf. A008290, A008291, A170942.

Programs

  • Maple
    a:=n->sum(n!*sum((-1)^k/(k-5)!, j=0..n), k=5..n): seq(-a(n)/6!, n=5..24);
restart: G(x):=exp(-x)/(1-x)*(x^6/6!): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=6..23); # Zerinvary Lajos, Apr 03 2009
  • Mathematica
    With[{nn=40}, Drop[CoefficientList[Series[Exp[-x]/(1 - x) x^6/6!, {x, 0, nn}], x]Range[0, nn]!, 6]] (* Vincenzo Librandi, Feb 19 2014 *)
  • PARI
    x='x+O('x^66); Vec( serlaplace(exp(-x)/(1-x)*(x^6/6!)) ) \\ Joerg Arndt, Feb 19 2014

Formula

a(n) = A008290(n,6).
E.g.f.: exp(-x)/(1-x)*(x^6/6!). [Zerinvary Lajos, Apr 03 2009]
O.g.f.: (1/6!)*Sum_{k>=6} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 15 2017
D-finite with recurrence +(-n+6)*a(n) +n*(n-7)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 06 2023

Extensions

Changed offset from 0 to 6 by Vincenzo Librandi, Feb 19 2014
Edited by Joerg Arndt, Feb 19 2014

A129149 Permutations with exactly 7 fixed points.

Original entry on oeis.org

1, 0, 36, 240, 2970, 34848, 454740, 6362928, 95450355, 1527194240, 25962321528, 467321755680, 8879113408308, 177582268088640, 3729227629977720, 82043007859339296, 1886989180765048965, 45287740338360829056
Offset: 7

Views

Author

Zerinvary Lajos, May 25 2007

Keywords

Links

Crossrefs

Cf. A008290, A008291, A170942.

Programs

  • Maple
    a:=n->sum(n!*sum((-1)^k/(k-6)!, j=0..n), k=6..n): seq(a(n)/7!, n=6..24);
restart: G(x):=exp(-x)/(1-x)*(x^7/7!): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=7..24); # Zerinvary Lajos, Apr 03 2009
  • Mathematica
    With[{nn=40}, Drop[CoefficientList[Series[Exp[-x]/(1 - x) x^7/7!, {x, 0, nn}], x]Range[0, nn]!, 7]] (* Vincenzo Librandi, Feb 19 2014 *)
  • PARI
    x='x+O('x^66); Vec( serlaplace(exp(-x)/(1-x)*(x^7/7!)) ) \\ Joerg Arndt, Feb 19 2014

Formula

a(n) = A008290(n,7).
E.g.f.: exp(-x)/(1-x)*(x^7/7!). [Zerinvary Lajos, Apr 03 2009]
Conjecture: (-n+7)*a(n) +n*(n-8)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 02 2015
O.g.f.: (1/7!)*Sum_{k>=7} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 15 2017
D-finite with recurrence (-n+7)*a(n) +n*(n-8)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 06 2023

Extensions

Changed offset from 0 to 7 by Vincenzo Librandi, Feb 19 2014
Edited by Joerg Arndt, Feb 19 2014

A129153 Rencontres numbers: permutations with exactly 8 fixed points.

Original entry on oeis.org

1, 0, 45, 330, 4455, 56628, 795795, 11930490, 190900710, 3245287760, 58415223438, 1109889169740, 22197783520770, 466153453732680, 10255375982438730, 235873647595600476, 5660967542295146895, 141524188557377590800
Offset: 8

Views

Author

Zerinvary Lajos, May 25 2007

Keywords

Links

Crossrefs

Column k=8 of A008290.
Cf. A008291, A170942.

Programs

  • Maple
    a:= n-> -sum((n-1)!*sum((-1)^k/(k-7)!, j=0..n-1), k=7..n-1)/8!: seq(a(n), n=8..30);
  • Mathematica
    With[{nn=40}, Drop[CoefficientList[Series[Exp[-x]/(1 - x) x^8/8!, {x, 0, nn}], x]Range[0, nn]!, 8]] (* Vincenzo Librandi, Feb 19 2014 *)
  • PARI
    x='x+O('x^66); Vec( serlaplace(exp(-x)/(1-x)*(x^8/8!)) ) \\ Joerg Arndt, Feb 19 2014

Formula

a(n) = A008290(n,8).
E.g.f.: exp(-x)/(1-x)*(x^8/8!). [Joerg Arndt, Feb 19 2014]
O.g.f.: (1/8!)*Sum_{k>=8} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 15 2017
D-finite with recurrence (-n+8)*a(n) +n*(n-9)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 06 2023

Extensions

Changed offset from 0 to 8 by Vincenzo Librandi, Feb 19 2014

A129217 Permutations with exactly 9 fixed points.

Original entry on oeis.org

1, 0, 55, 440, 6435, 88088, 1326325, 21209760, 360590230, 6490575520, 123321027258, 2466420377200, 51794828215130, 1139486220235440, 26208183066232310, 628996393588267936, 15724909839708741375
Offset: 9

Views

Author

Zerinvary Lajos, May 25 2007

Keywords

Links

Crossrefs

Cf. A008290, A008291, A170942.

Programs

  • Maple
    a:=n->sum(n!*sum((-1)^k/(k-8)!, j=0..n), k=8..n): seq(a(n)/9!, n=8..27);
restart: G(x):=exp(-x)/(1-x)*(x^9/9!): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=9..25); # Zerinvary Lajos, Apr 03 2009
  • Mathematica
    With[{nn=40}, Drop[CoefficientList[Series[Exp[-x]/(1 - x) x^9/9!, {x, 0, nn}], x]Range[0, nn]!, 9]] (* Vincenzo Librandi, Feb 19 2014 *)
  • PARI
    x='x+O('x^66); Vec( serlaplace(exp(-x)/(1-x)*(x^9/9!)) ) \\ Joerg Arndt, Feb 19 2014

Formula

a(n) = A008290(n,9).
E.g.f.: exp(-x)/(1-x)*(x^9/9!). [Zerinvary Lajos, Apr 03 2009]
O.g.f.: (1/9!)*Sum_{k>=9} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 15 2017

Extensions

Changed offset from 0 to 9 by Vincenzo Librandi, Feb 19 2014
Edited by Joerg Arndt, Feb 19 2014

A129238 Permutations with exactly 11 fixed points.

Original entry on oeis.org

1, 0, 78, 728, 12285, 192192, 3279640, 59001696, 1121107806, 22421988160, 470862104076, 10358965584240, 238256209789598, 5718149032454208, 142953725815812600, 3716796871203401440, 100353515522504876775
Offset: 11

Views

Author

Zerinvary Lajos, May 25 2007

Keywords

Links

Crossrefs

Cf. A008290, A008291, A170942.

Programs

  • Maple
    a:=n->sum(n!*sum((-1)^k/(k-10)!, j=0..n), k=10..n): seq(a(n)/11!, n=10..27);
restart: G(x):=exp(-x)/(1-x)*(x^11/11!): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=11..27); # Zerinvary Lajos, Apr 03 2009
  • Mathematica
    With[{nn=40}, Drop[CoefficientList[Series[Exp[-x]/(1 - x) x^11/11!, {x, 0, nn}], x]Range[0, nn]!, 11]] (* Vincenzo Librandi, Feb 19 2014 *)
  • PARI
    x='x+O('x^66); Vec( serlaplace(exp(-x)/(1-x)*(x^11/11!)) ) \\ Joerg Arndt, Feb 19 2014

Formula

a(n) = A008290(n,11).
E.g.f.: exp(-x)/(1-x)*(x^11/11!) . [Zerinvary Lajos, Apr 03 2009]
O.g.f.: (1/11!)*Sum_{k>=11} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 15 2017

Extensions

Changed offset from 0 to 11 by Vincenzo Librandi, Feb 19 2014
Edited by Joerg Arndt, Feb 19 2014

A129255 Permutations with exactly 12 fixed points.

Original entry on oeis.org

1, 0, 91, 910, 16380, 272272, 4919460, 93419352, 1868513010, 39238479280, 863247190806, 19854684036460, 476512419579196, 11912810484279600, 309733072600927300, 8362792960207653240, 234158202885844712475
Offset: 12

Views

Author

Zerinvary Lajos, May 25 2007

Keywords

Links

Crossrefs

Column k=12 of A008290.
Cf. A008291, A170942.

Programs

  • Maple
    a:=n->sum(n!*sum((-1)^k/(k-11)!, j=0..n), k=11..n): seq(-a(n)/12!, n=11..28);
restart: G(x):=exp(-x)/(1-x)*(x^12/12!): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=12..28);# Zerinvary Lajos, Apr 03 2009
  • Mathematica
    With[{nn=40}, Drop[CoefficientList[Series[Exp[-x]/(1 - x) x^12/12!, {x, 0, nn}], x]Range[0, nn]!, 12]] (* Vincenzo Librandi, Feb 19 2014 *)
  • PARI
    x='x+O('x^66); Vec( serlaplace(exp(-x)/(1-x)*(x^12/12!)) ) \\ Joerg Arndt, Feb 19 2014

Formula

E.g.f.: exp(-x)/(1-x)*(x^12/12!). [Zerinvary Lajos, Apr 03 2009]
O.g.f.: (1/12!)*Sum_{k>=12} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 15 2017

Extensions

Changed offset from 0 to 12 by Vincenzo Librandi, Feb 19 2014
Edited by Joerg Arndt, Feb 19 2014
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