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.

Showing 1-8 of 8 results.

A053102 a(n) = ((6*n+9)(!^6))/9(!^6), related to A034723 (((6*n+3)(!^6))/3 sextic, or 6-factorials).

Original entry on oeis.org

1, 15, 315, 8505, 280665, 10945935, 492567075, 25120920825, 1431892487025, 90209226682575, 6224436641097675, 466832748082325625, 37813452594668375625, 3289770375736148679375, 305948644943461827181875
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Row m=9 of the array A(7; m,n) := ((6*n+m)(!^6))/m(!^6), m >= 0, n >= 0.

Links

Crossrefs

Cf. A047058, A008542(n+1), A034689(n+1), A034723(n+1), A034724(n+1), A034787(n+1), A034788(n+1), A053100, A053101, this sequence, A053103 (rows m=0..10).

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-6*x)^(15/6))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018
  • Mathematica
    s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 14, 5!, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 6*x)^(15/6), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(1/(1-6*x)^(15/6))) \\ G. C. Greubel, Aug 15 2018

Formula

a(n) = ((6*n+9)(!^6))/9(!^6) = A034723(n+2)/9.
E.g.f.: 1/(1-6*x)^(15/6).

A008542 Sextuple factorial numbers: Product_{k=0..n-1} (6*k+1).

Original entry on oeis.org

1, 1, 7, 91, 1729, 43225, 1339975, 49579075, 2131900225, 104463111025, 5745471106375, 350473737488875, 23481740411754625, 1714167050058087625, 135419196954588922375, 11510631741140058401875, 1047467488443745314570625, 101604346379043295513350625
Offset: 0

Views

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

Comments

a(n), n>=1, enumerates increasing heptic (7-ary) trees with n vertices. - Wolfdieter Lang, Sep 14 2007; see a D. Callan comment on A007559 (number of increasing quarterny trees).

Links

Crossrefs

Cf. A085158, A034689, A034723, A034724, A034787, A034788, A004993, A047058, A047657, A051151.
Cf. k-fold factorials: A000142, A001147 (and A000165, A006882), A007559 (and A032031, A008544, A007661), A007696 (and A001813, A008545, A047053, A007662), A008548 (and A052562, A047055, A085157), A045754 (and A084947, A114799), A045755.

Programs

  • GAP
    List([0..20], n-> Product([0..n-1], k-> (6*k+1) )); # G. C. Greubel, Aug 17 2019
  • Magma
    [1] cat [(&*[(6*k+1): k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 17 2019
  • Maple
    a := n -> mul(6*k+1, k=0..n-1);
G(x):=(1-6*x)^(-1/6): 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=0..15); # Zerinvary Lajos, Apr 03 2009
  • Mathematica
    Table[Product[(6*k+1), {k,0,n-1}], {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008, modified by G. C. Greubel, Aug 17 2019 *)
FoldList[Times, 1, 6Range[0, 20] + 1] (* Vincenzo Librandi, Jun 10 2013 *)
Table[6^n*Pochhammer[1/6, n], {n,0,20}] (* G. C. Greubel, Aug 17 2019 *)
  • PARI
    a(n)=prod(k=1,n-1,6*k+1) \\ Charles R Greathouse IV, Jul 19 2011
  • Sage
    [product((6*k+1) for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 17 2019

Formula

E.g.f.: (1-6*x)^(-1/6).
a(n) ~ 2^(1/2)*Pi^(1/2)*Gamma(1/6)^-1*n^(-1/3)*6^n*e^-n*n^n*{1 + 1/72*n^-1 - ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
a(n) = Sum_{k=0..n} (-6)^(n-k)*A048994(n, k). - Philippe Deléham, Oct 29 2005
G.f.: 1+x/(1-7x/(1-6x/(1-13x/(1-12x/(1-19x/(1-18x/(1-25x/(1-24x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-5)^n*Sum_{k=0..n} (6/5)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/Q(0) where Q(k) = 1 - x*(6*k+1)/(1 - x*(6*k+6)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
a(n) = A085158(6*n-5). - M. F. Hasler, Feb 23 2018
D-finite with recurrence: a(n) +(-6*n+5)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 1 + (e/6^5)^(1/6)*(Gamma(1/6) - Gamma(1/6, 1/6)). - Amiram Eldar, Dec 18 2022

A034689 a(n) = n-th sextic factorial number divided by 2.

Original entry on oeis.org

1, 8, 112, 2240, 58240, 1863680, 70819840, 3116072960, 155803648000, 8725004288000, 540950265856000, 36784618078208000, 2722061737787392000, 217764939022991360000, 18727784755977256960000, 1722956197549907640320000, 168849707359890948751360000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A007559, A008542, A034723, A034724, A034787, A034788, A047657, A047058.

Programs

  • Magma
    [n le 1 select 1 else (6*n-4)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 21 2022
  • Mathematica
    Table[6^n*Pochhammer[1/3, n]/2, {n, 40}] (* G. C. Greubel, Oct 21 2022 *)
  • SageMath
    [6^n*rising_factorial(1/3,n)/2 for n in range(1,40)] # G. C. Greubel, Oct 21 2022

Formula

2*a(n) = (6*n-4)(!^6) = Product_{j=1..n} (6*j-4) = 2^n*A007559(n), A007559(n) = (3*n-2)(!^3) = Product_{j=1..n} (3*j-2).
E.g.f.: (-1 + (1-6*x)^(-1/3))/2.
D-finite with recurrence: a(n) = 2*(3*n-2)*a(n-1). - R. J. Mathar, Feb 24 2020
a(n) = 3*6^(n-1)*Pochhammer(n, 1/3). - G. C. Greubel, Oct 21 2022
From Amiram Eldar, Dec 18 2022: (Start)
a(n) = A047657(n)/2.
Sum_{n>=1} 1/a(n) = 2*(e/6^4)^(1/6)*(Gamma(1/3, 1/6) - Gamma(1/3)). (End)

A034787 a(n) = n-th sextic factorial number divided by 5.

Original entry on oeis.org

1, 11, 187, 4301, 124729, 4365515, 178986115, 8412347405, 445854412465, 26305410335435, 1709851671803275, 121399468698032525, 9347759089748504425, 775864004449125867275, 69051896395972202187475, 6559930157617359207810125, 662552945919353279988822625
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A008542, A034689, A034723, A034724, A034788.

Programs

  • GAP
    List([1..20], n-> Product([1..n], j-> 6*j-1)/5 ); # G. C. Greubel, Nov 11 2019
  • Magma
    [(&*[6*j-1: j in [1..n]])/5: n in [1..20]]; // G. C. Greubel, Nov 11 2019
  • Maple
    seq( mul(6*j-1, j=1..n)/5, n=1..20); # G. C. Greubel, Nov 11 2019
  • Mathematica
    Table[6^n*Pochhammer[5/6, n]/5, {n,20}] (* G. C. Greubel, Nov 11 2019 *)
With[{nn=20},CoefficientList[Series[(-1+(1-6x)^(-5/6))/5,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 21 2024 *)
  • PARI
    vector(20, n, prod(j=1,n, 6*j-1)/5 ) \\ G. C. Greubel, Nov 11 2019
  • Sage
    [product( (6*j-1) for j in (1..n))/5 for n in (1..20)] # G. C. Greubel, Nov 11 2019

Formula

5*a(n) = (6*n-1)(!^6) = Product_{j=1..n} (6*j-1) = (6*n)!/(3^(2*n)*2^(2*n+1)*(2*n)!*A008542(n)*A007559(n)*A034000(n)).
E.g.f.: (-1 + (1-6*x)^(-5/6))/5.
a(n+1) ~ sqrt(2*Pi) * 6/(5*Gamma(5/6)) * n^(4/3) * (6*n/e)^n * (1 + (61/72)/n + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
D-finite with recurrence: a(n) +(-6*n+1)*a(n-1)=0. - R. J. Mathar, Feb 24 2020
Sum_{n>=1} 1/a(n) = 5*(e/6)^(1/6)*(Gamma(5/6) - Gamma(5/6, 1/6)). - Amiram Eldar, Dec 18 2022

A034788 a(n) is the n-th sextic factorial number divided by 6.

Original entry on oeis.org

1, 12, 216, 5184, 155520, 5598720, 235146240, 11287019520, 609499054080, 36569943244800, 2413616254156800, 173780370299289600, 13554868883344588800, 1138608986200945459200, 102474808758085091328000, 9837581640776168767488000, 1003433327359169214283776000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A008542, A034689, A034723, A034724, A034787.

Programs

  • GAP
    List([1..20], n-> 6^(n-1)*Factorial(n) ); # G. C. Greubel, Nov 11 2019
  • Magma
    [6^(n-1)*Factorial(n): n in [1..20]]; // G. C. Greubel, Nov 11 2019
  • Maple
    seq(6^(n-1)*n!, n=1..20); # G. C. Greubel, Nov 11 2019
  • Mathematica
    Table[6^(n-1)*n!,{n,20}] (* Harvey P. Dale, Dec 22 2013 *)
  • PARI
    vector(20, n, 6^(n-1)*n!) \\ G. C. Greubel, Nov 11 2019
  • Sage
    [6^(n-1)*factorial(n) for n in (1..20)] # G. C. Greubel, Nov 11 2019

Formula

6*a(n) = (6*n)(!^6) = Product_{j=1..n} 6*j = 6^n*n!.
E.g.f.: (-1 + 1/(1-6*x))/6.
D-finite with recurrence: a(n) - 6*n*a(n-1) = 0. - R. J. Mathar, Feb 24 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 6*(exp(1/6)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 6*(1-exp(-1/6)). (End)

A053101 a(n) = ((6*n+8)(!^6))/8(!^6), related to A034689 (((6*n+2)(!^6))/2 sextic, or 6-factorials).

Original entry on oeis.org

1, 14, 280, 7280, 232960, 8852480, 389509120, 19475456000, 1090625536000, 67618783232000, 4598077259776000, 340257717223424000, 27220617377873920000, 2340973094497157120000, 215369524693738455040000
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Row m=7 of the array A(7; m,n) := ((6*n+m)(!^6))/m(!^6), m >= 0, n >= 0.

Links

Crossrefs

Cf. A047058, A008542(n+1), A034689(n+1), A034723(n+1), A034724(n+1), A034787(n+1), A034788(n+1), A053100, this sequence, A053102, A053103 (rows m=0..10).

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-6*x)^(7/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018
  • Mathematica
    s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 13, 5!, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 6*x)^(7/3), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(1/(1-6*x)^(7/3))) \\ G. C. Greubel, Aug 15 2018

Formula

a(n) = ((6*n+8)(!^6))/8(!^6)= A034689(n+2)/8.
E.g.f.: 1/(1-6*x)^(7/3).

A053103 a(n) = ((6*n+10)(!^6))/10(!^6), related to A034724 (((6*n+4)(!^6))/4 sextic, or 6-factorials).

Original entry on oeis.org

1, 16, 352, 9856, 335104, 13404160, 616591360, 32062750720, 1859639541760, 119016930672640, 8331185147084800, 633170071178444800, 51919945836632473600, 4568955233623657676800, 429481791960623821619200
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Row m=10 of the array A(7; m,n) := ((6*n+m)(!^6))/m(!^6), m >= 0, n >= 0.

Links

Crossrefs

Cf. A047058, A008542(n+1), A034689(n+1), A034723(n+1), A034724(n+1), A034787(n+1), A034788(n+1), A053100, A053101, A053102, this sequence (rows m=0..10).

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-6*x)^(8/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 16 2018
  • Mathematica
    s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 15, 5!, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 6*x)^(16/6), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 16 2018 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(1/(1-6*x)^(8/3))) \\ G. C. Greubel, Aug 16 2018

Formula

a(n) = ((6*n+10)(!^6))/10(!^6) = A034724(n+2)/10.
E.g.f.: 1/(1-6*x)^(8/3).

A053100 a(n) = ((6*n+7)(!^6))/7, related to A008542 ((6*n+1)(!^6) sextic, or 6-factorials).

Original entry on oeis.org

1, 13, 247, 6175, 191425, 7082725, 304557175, 14923301575, 820781586625, 50067676784125, 3354534344536375, 244881007151155375, 19345599564941274625, 1644375963020008343125, 149638212634820759224375
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Row m=7 of the array A(7; m,n) := ((6*n+m)(!^6))/m(!^6), m >= 0, n >= 0.

Links

Crossrefs

Cf. A047058, A008542(n+1), A034689(n+1), A034723(n+1), A034724(n+1), A034787(n+1), A034788(n+1), this sequence, A053101, A053102, A053103 (rows m=0..10).

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-6*x)^(13/6))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018
  • Mathematica
    s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 12, 5!, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn=20},CoefficientList[Series[1/(1-6x)^(13/6),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 20 2015 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(1/(1-6*x)^(13/6))) \\ G. C. Greubel, Aug 15 2018

Formula

a(n) = ((6*n+7)(!^6))/7(!^6) = A008542(n+2)/7.
E.g.f.: 1/(1-6*x)^(13/6).
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.