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 1-10 of 11 results. Next

A051688 a(n) = (5*n+7)(!^5)/7(!^5), related to A034323 ((5*n+2)(!^5) quintic, or 5-factorials).

Original entry on oeis.org

1, 12, 204, 4488, 121176, 3877632, 143472384, 6025840128, 283214486016, 14727153272832, 839447736551424, 52045759666188288, 3487065897634615296, 251068744629692301312, 19332293336486307201024
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Row m=7 of the array A(6; m,n) := ((5*n+m)(!^5))/m(!^5), m >= 0, n >= 0.

Links

Crossrefs

Cf. A052562, A008548(n+1), A034323(n+1), A034300(n+1), A034301(n+1), A034325(n+1), A051687-A051691 (rows m=0..10).

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-5*x)^(12/5))); [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, 11, 5!, 5}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 5*x)^(12/5), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(1/(1-5*x)^(12/5))) \\ G. C. Greubel, Aug 15 2018

Formula

a(n) = ((5*n+7)(!^5))/7(!^5) = A034323(n+2)/7.
E.g.f.: 1/(1-5*x)^(12/5).

A034301 a(n) = n-th quintic factorial number divided by 4.

Original entry on oeis.org

1, 9, 126, 2394, 57456, 1666224, 56651616, 2209413024, 97214173056, 4763494479744, 257228701906176, 15176493412464384, 971295578397720576, 67019394909442719744, 4959435223298761261056, 391795382640602139623424, 32910812141810579728367616, 2929062280621141595824717824
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A008546, A008548, A034300, A025750.

Programs

  • GAP
    List([1..20], n-> Product([1..n], k-> 5*k-1)/4 ); # G. C. Greubel, Aug 23 2019
  • Magma
    [&*[5*k-1: k in [1..n]]/4: n in [1..20]]; // G. C. Greubel, Aug 23 2019
  • Maple
    a:= n-> mul(5*k-1, k=1..n)/4: seq(a(n), n=1..20); # G. C. Greubel, Aug 23 2019
  • Mathematica
    Table[-5^(n+1)*Pochhammer[-1/5, n+1]/4, {n,20}] (* G. C. Greubel, Aug 23 2019 *)
  • PARI
    a(n) = prod(k=1,n, 5*k-1)/4;
vector(20, n, a(n)) \\ G. C. Greubel, Aug 23 2019
  • Sage
    [-5^(n+1)*rising_factorial(-1/5, n+1)/4 for n in (1..20)] # G. C. Greubel, Aug 23 2019

Formula

a(n) = A008546(n)/4.
4*a(n) = (5*n-1)(!^5) = Product_{j=1..n} (5*j-1).
a(n) = (5*n)!/(5^n*n!*A008548(n)*2*A034323(n)*3*A034300(n)).
E.g.f.: (-1 + (1-5*x)^(-4/5))/4, a(0) = 0.
a(n) ~ sqrt(2*Pi) * 5/(4*Gamma(4/5)) * n^(13/10) * (5*n/e)^n * (1 + (241/300)/n + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
D-finite with recurrence: a(n) +(-5*n+1)*a(n-1)=0. - R. J. Mathar, Feb 20 2020
Sum_{n>=1} 1/a(n) = 4*(e/5)^(1/5)*(Gamma(4/5) - Gamma(4/5, 1/5)). - Amiram Eldar, Dec 19 2022

Extensions

Terms a(17) onward added by G. C. Greubel, Aug 23 2019

A034325 a(n) is the n-th quintic factorial number divided by 5.

Original entry on oeis.org

1, 10, 150, 3000, 75000, 2250000, 78750000, 3150000000, 141750000000, 7087500000000, 389812500000000, 23388750000000000, 1520268750000000000, 106418812500000000000, 7981410937500000000000, 638512875000000000000000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A008548, A034300, A034301, A034323, A052562.

Programs

  • GAP
    List([1..20], n-> 5^(n-1)*Factorial(n) ); # G. C. Greubel, Aug 23 2019
  • Magma
    [5^(n-1)*Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 23 2019
  • Maple
    seq(5^(n-1)*n!, n=1..20); # G. C. Greubel, Aug 23 2019
  • Mathematica
    Array[5^(# - 1) #! &, 16] (* Michael De Vlieger, May 30 2019 *)
  • PARI
    vector(20, n, 5^(n-1)*n!) \\ G. C. Greubel, Aug 23 2019
  • Sage
    [5^(n-1)*factorial(n) for n in (1..20)] # G. C. Greubel, Aug 23 2019

Formula

5*a(n) = (5*n)(!^5) = Product_{j=1..n} 5*j = 5^(n-1)*n!.
E.g.f.: (-1 + (1-5*x)^(-1))/5, a(0) = 0.
D-finite with recurrence: a(n) - 5*n*a(n-1) = 0. - R. J. Mathar, Feb 24 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 5*(exp(1/5)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*(1-exp(-1/5)). (End)

A035273 One quarter of deca-factorial numbers.

Original entry on oeis.org

1, 14, 336, 11424, 502656, 27143424, 1737179136, 128551256064, 10798305509376, 1015040717881344, 105564234659659776, 12034322751201214464, 1492256021148950593536, 199962306833959379533824, 28794572184090150652870656, 4434364116349883200542081024
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A034323, A035272, A035274, A035275, A035276, A035277, A035278, A035279, A045757.

Programs

  • GAP
    List([1..20], n-> Product([1..n], j-> 10*j-6)/4 ); # G. C. Greubel, Nov 11 2019
  • Magma
    [(&*[10*j-6: j in [1..n]])/4: n in [1..20]]; // G. C. Greubel, Nov 11 2019
  • Maple
    seq( mul(10*j-6, j=1..n)/4, n=1..20); # G. C. Greubel, Nov 11 2019
  • Mathematica
    Table[10^n*Pochhammer[4/10, n]/4, {n,20}] (* G. C. Greubel, Nov 11 2019 *)
  • PARI
    vector(20, n, prod(j=1,n, 10*j-6)/4 ) \\ G. C. Greubel, Nov 11 2019
  • Sage
    [product( (10*j-6) for j in (1..n))/4 for n in (1..20)] # G. C. Greubel, Nov 11 2019

Formula

4*a(n) = (10*n-6)(!^10) = Product_{j=1..n} (10*j-6).
a(n) = 2^(n+1)*A034323(n) where 2*A034323(n)= (5*n-3)(!^5) = Product_{j=1..n} (5*j-3).
E.g.f.: (-1 + (1-10*x)^(-2/5))/4.
a(n) = (Pochhammer(4/10,n)*10^n)/4.
Sum_{n>=1} 1/a(n) = 4*(e/10^6)^(1/10)*(Gamma(2/5) - Gamma(2/5, 1/10)). - Amiram Eldar, Dec 22 2022

A051687 a(n) = (5*n+6)(!^5)/6, related to A008548 ((5*n+1)(!^5) quintic, or 5-factorials).

Original entry on oeis.org

1, 11, 176, 3696, 96096, 2978976, 107243136, 4396968576, 202260554496, 10315288279296, 577656143640576, 35237024762075136, 2325643634296958976, 165120698035084087296, 12549173050666390634496
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Row m=6 of the array A(6; m,n) := ((5*n+m)(!^5))/m(!^5), m >= 0, n >= 0.

Links

Crossrefs

Cf. A052562, A008548(n+1), A034323(n+1), A034300(n+1), A034301(n+1), A034325(n+1), A051687-A051691 (rows m=0..10).

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-5*x)^(11/5))); [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, 10, 5!, 5}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 5*x)^(11/5), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(1/(1-5*x)^(11/5))) \\ G. C. Greubel, Aug 15 2018

Formula

a(n) = ((5*n+6)(!^5))/6(!^5).
E.g.f.: 1/(1-5*x)^(11/5).

A051691 a(n) = (5*n+10)(!^5)/10(!^5), related to A052562 ((5*n)(!^5) quintic, or 5-factorials).

Original entry on oeis.org

1, 15, 300, 7500, 225000, 7875000, 315000000, 14175000000, 708750000000, 38981250000000, 2338875000000000, 152026875000000000, 10641881250000000000, 798141093750000000000, 63851287500000000000000
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Row m=10 of the array A(6; m,n) := ((5*n+m)(!^5))/m(!^5), m >= 0, n >= 0.

Links

Crossrefs

Cf. A052562, A008548(n+1), A034323(n+1), A034300(n+1), A034301(n+1), A034325(n+1), A051687-A051691 (rows m=0..10).

Programs

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

Formula

a(n) = ((5*n+10)(!^5))/10(!^5) = A052562(n+2)/(5*10).
E.g.f.: 1/(1-5*x)^3.

A051690 a(n) = (5*n+9)(!^5)/9(!^5), related to A034301 ((5*n+2)(!^5) quintic, or 5-factorials).

Original entry on oeis.org

1, 14, 266, 6384, 185136, 6294624, 245490336, 10801574784, 529277164416, 28580966878464, 1686277045829376, 107921730933080064, 7446599434382524416, 551048358144306806784, 43532820293400237735936, 3656756904645619969818624, 325451364513460177313857536
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Row m=9 of the array A(6; m,n) := ((5*n+m)(!^5))/m(!^5), m >= 0, n >= 0.

Links

Crossrefs

Cf. A052562, A008548(n+1), A034323(n+1), A034300(n+1), A034301(n+1), A034325(n+1), A051687-A051691 (rows m=0..10).

Programs

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

Formula

a(n) = ((5*n+9)(!^5))/9(!^5) = A034301(n+2)/9.
E.g.f.: 1/(1-5*x)^(14/5).

A303488 a(n) = n! * [x^n] 1/(1 - 5*x)^(n/5).

Original entry on oeis.org

1, 1, 14, 312, 9576, 375000, 17873856, 1004306688, 65006637696, 4763494479744, 389812500000000, 35237024762075136, 3487065897634615296, 374960171943074285568, 43532820293400237735936, 5427359437500000000000000, 723181462895975365595529216, 102563963819340862347122245632
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 24 2018

Keywords

Examples

			a(1) = 1;
a(2) = 2*7 = 14;
a(3) = 3*8*13 = 312;
a(4) = 4*9*14*19 = 9576;
a(5) = 5*10*15*20*25 = 375000, etc.

Links

Crossrefs

Column k=5 of A303489.
Cf. A008546, A008548, A034300, A034301, A034323, A034325, A047055, A047056, A051687, A051688, A051689, A051690, A051691, A052562, A113551, A303486, A303487.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[1/(1 - 5 x)^(n/5), {x, 0, n}], {n, 0, 17}]
Table[Product[5 k + n, {k, 0, n - 1}], {n, 0, 17}]
Table[5^n Pochhammer[n/5, n], {n, 0, 17}]

Formula

a(n) = Product_{k=0..n-1} (5*k + n).
a(n) = 5^n*Gamma(6*n/5)/Gamma(n/5).
a(n) ~ 6^(6*n/5-1/2)*n^n/exp(n).

A051689 a(n) = (5*n+8)(!^5)/8(!^5), related to A034300 ((5*n+3)(!^5) quintic, or 5-factorials).

Original entry on oeis.org

1, 13, 234, 5382, 150696, 4972968, 188972784, 8125829712, 390039826176, 20672110787328, 1198982425665024, 75535892816896512, 5136440711548962816, 374960171943074285568, 29246893411559794274304
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Row m=8 of the array A(6; m,n) := ((5*n+m)(!^5))/m(!^5), m >= 0, n >= 0.

Links

Crossrefs

Cf. A052562, A008548(n+1), A034323(n+1), A034300(n+1), A034301(n+1), A034325(n+1), A051687-A051691 (rows m=0..10).

Programs

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

Formula

a(n) = ((5*n+8)(!^5))/8(!^5) = A034300(n+2)/8.
E.g.f.: 1/(1-5*x)^(13/5).

A081408 a(n) = (n+1)*a(n-5), with a(0)=a(1)=a(2)=a(3)=a(4)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 7, 8, 9, 10, 66, 84, 104, 126, 150, 1056, 1428, 1872, 2394, 3000, 22176, 31416, 43056, 57456, 75000, 576576, 848232, 1205568, 1666224, 2250000, 17873856, 27143424, 39783744, 56651616, 78750000, 643458816, 1004306688, 1511782272
Offset: 0

Views

Author

Labos Elemer, Apr 01 2003

Keywords

Comments

Quintic factorial sequences are generated by single 5-order recursion and appear in unified form.

Examples

			A008548, A034323, A034300, A034301, A034325 sequences are combed together as A081408(5n+r) with r=0,1,2,3,4.

Links

Crossrefs

Cf. A001147, A002866, A034001, A007599, A034000, A007696, A000407, A034176, A034177, A008548, A034323, A034300, A034301, A034325 [double, triple, quartic, quintic, factorial subsequences], generated together in A081405-A081408.

Programs

  • GAP
    a:=[1,1,1,1,1];; for n in [6..40] do a[n]:=n*a[n-5]; od; a; # G. C. Greubel, Aug 15 2019
  • Haskell
    a081407 n = a081408_list !! n
a081407_list = 1 : 1 : 1 : 1 : zipWith (*) [5..] a081407_list
-- Reinhard Zumkeller, Jan 05 2012
  • Magma
    [n le 5 select 1 else n*Self(n-5): n in [1..40]]; // G. C. Greubel, Aug 15 2019
  • Mathematica
    a[0]=a[1]=a[2]=a[3]=a[4]=1; a[x_]:= (x+1)*a[x-5]; Table[a[n], {n, 40}]
  • PARI
    m=30; v=concat([1,1,1,1,1], vector(m-5)); for(n=6, m, v[n]=n*v[n-5] ); v \\ G. C. Greubel, Aug 15 2019
  • Sage
    def a(n):
    if (n<5): return 1
    else: return (n+1)*a(n-5)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 15 2019
Showing 1-10 of 11 results. Next

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