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.

A049209 a(n) = -Product_{k=0..n} (7*k-1); sept-factorial numbers.

Original entry on oeis.org

1, 6, 78, 1560, 42120, 1432080, 58715280, 2818333440, 155008339200, 9610517030400, 663125675097600, 50397551307417600, 4182996758515660800, 376469708266409472000, 36517561701841718784000, 3797826416991538753536000, 421558732286060801642496000
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A034833, A045754, A048994, A051188, A084947, A144739, A144827, A147585.
Row sums of triangle A051186 (scaled Stirling1 triangle).
Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), this sequence (m=7), A049210 (m=8), A049211 (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).

Programs

  • Magma
    [ -&*[ (7*k-1): k in [0..n-1] ]: n in [1..15] ]; // Klaus Brockhaus, Nov 10 2008
  • Mathematica
    CoefficientList[Series[(1-7*x)^(-6/7),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)
With[{m=7}, Table[m^n*Pochhammer[(m-1)/m, n], {n, 0, 30}]] (* G. C. Greubel, Feb 16 2022 *)
  • Sage
    m=7; [m^n*rising_factorial((m-1)/m, n) for n in (0..30)] # G. C. Greubel, Feb 16 2022

Formula

a(n) = 6*A034833(n) = (7*n-1)*(!^7), n >= 1, a(0) := 1.
a(n) = Product_{k=1..n} (7*k - 1). a(0) = 1; a(n) = (7*n - 1)*a(n-1) for n > 0. - Klaus Brockhaus, Nov 10 2008
G.f.: 1/(1-6*x/(1-7*x/(1-13*x/(1-14*x/(1-20*x/(1-21*x/(1-27*x/(1-28*x/(1-...(continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-1)^n*Sum_{k=0..n} 7^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
a(n) = 7^n * Gamma(n+6/7) / Gamma(6/7). - Vaclav Kotesovec, Jan 28 2015
E.g.f.: (1-7*x)^(-6/7). - Vaclav Kotesovec, Jan 28 2015
From Nikolaos Pantelidis, Dec 19 2020: (Start)
G.f.: 1/G(0) where G(k) = 1 - (14*k+6)*x - 7*(k+1)*(7*k+6)*x^2/G(k+1); (continued fraction).
which starts as 1/(1-6*x-42*x^2/(1-20*x-182*x^2/(1-34*x-420*x^2/(1-48*x-756*x^2/(1-62*x-1190*x^2/(1-... )))))) (Jacobi continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - (7*k+6)*x/(1 - (7*k+7)*x/Q(k+1) ); (continued fraction). (End)
Sum_{n>=0} 1/a(n) = 1 + (e/7)^(1/7)*(Gamma(6/7) - Gamma(6/7, 1/7)). - Amiram Eldar, Dec 19 2022

A034834 One seventh of sept-factorial numbers.

Original entry on oeis.org

1, 14, 294, 8232, 288120, 12101040, 592950960, 33205253760, 2091930986880, 146435169081600, 11275508019283200, 947142673619788800, 86189983299400780800, 8446618363341276518400, 886894928150834034432000, 99332231952893411856384000, 11820535602394316010909696000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A045754, A034829, A034830, A034831, A034832, A034833, A051188.

Programs

  • Magma
    [7^(n-1)*Factorial(n): n in [1..30]]; // G. C. Greubel, Feb 22 2018
  • Mathematica
    Table[7^(n-1)*n!, {n,1,30}] (* or *) Drop[With[{nn = 50},CoefficientList[ Series[x/(1-7*x), {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 22 2018 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(x/(1-7*x))) \\ G. C. Greubel, Feb 22 2018

Formula

7*a(n) = (7*n)(!^7) = Product_{j=1..n} 7*j = 7^n*n!.
E.g.f.: x/(1-7*x).
a(n) = A051188(n)/7.
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 7*(exp(1/7)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 7*(1-exp(-1/7)). (End)

Extensions

More terms from G. C. Greubel, Feb 22 2018

A034835 Expansion of 1/(1-49*x)^(1/7); related to sept-factorial numbers A045754.

Original entry on oeis.org

1, 7, 196, 6860, 264110, 10722866, 450360372, 19365495996, 847240449825, 37560993275575, 1682732498745760, 76028913806967520, 3459315578217022160, 158330213003009860400, 7283189798138453578400, 336483368673996555322080, 15604416222256590253061460, 726064307753233111186565580
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A045754, A004993, A034829, A034830, A034831, A034832, A034833, A034834, A220086.

Programs

  • Magma
    Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!(1/(1 - 49*x)^(1/7))); // G. C. Greubel, Feb 22 2018
  • Mathematica
    CoefficientList[Series[1/(1 - 49*x)^(1/7), {x,0,50}], x] (* G. C. Greubel, Feb 22 2018 *)
  • PARI
    my(x='x+O('x^30)); Vec(1/(1 - 49*x)^(1/7)) \\ G. C. Greubel, Feb 22 2018

Formula

a(n) = 7^n*A045754(n)/n!, n >= 1, where A045754(n) = (7*n-6)(!^7) = Product_{j=1..n} (7*j-6).
G.f.: (1-49*x)^(-1/7).
D-finite with recurrence: n*a(n) + 7*(-7*n+6)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
a(n) ~ 7^(2*n) * n^(-6/7) / Gamma(1/7). - Amiram Eldar, Aug 18 2025

A034829 a(n) = n-th sept-factorial number divided by 2.

Original entry on oeis.org

1, 9, 144, 3312, 99360, 3676320, 161758080, 8249662080, 478480400640, 31101226041600, 2239288274995200, 176903773724620800, 15213724540317388800, 1414876382249517158400, 141487638224951715840000, 15139177290069833594880000, 1725866211067961029816320000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A045754, A034830, A034831, A034832, A034833, A034834, A084947.

Programs

  • Mathematica
    Drop[With[{nn = 50}, CoefficientList[Series[(-1 + (1 - 7*x)^(-2/7))/2, {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 23 2018 *)
  • PARI
    vector(20, n, prod(j=1, n, 7*j-5)/2) \\ Michel Marcus, Jan 07 2015

Formula

2*a(n) = (7*n-5)(!^7) = Product_{j=1..n} (7*j-5).
E.g.f.: (-1 + (1-7*x)^(-2/7))/2.
D-finite with recurrence: a(n) +(-7*n+5)*a(n-1)=0. - R. J. Mathar, Feb 24 2020
From Amiram Eldar, Dec 19 2022: (Start)
a(n) = A084947(n)/2.
Sum_{n>=1} 1/a(n) = 2*(e/7^5)^(1/7)*(Gamma(2/7) - Gamma(2/7, 1/7)). (End)

A034830 a(n) = n-th sept-factorial number divided by 3.

Original entry on oeis.org

1, 10, 170, 4080, 126480, 4806240, 216280800, 11246601600, 663549494400, 43794266630400, 3196981464019200, 255758517121536000, 22250990989573632000, 2091593153019921408000, 211250908455012062208000, 22815098113141302718464000, 2623736283011249812623360000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A045754, A034829, A034831, A034832, A034833, A034834, A144739.

Programs

  • Mathematica
    Drop[With[{nn = 40}, CoefficientList[Series[(-1 + (1 - 7*x)^(-3/7))/3, {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 23 2018 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace((-1 + (1-7*x)^(-3/7))/3)) \\ G. C. Greubel, Feb 23 2018

Formula

3*a(n) = (7*n-4)(!^7) = Product_{j=1..n} (7*j-4).
E.g.f.: (-1 + (1-7*x)^(-3/7))/3.
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A144739(n)/3.
Sum_{n>=1} 1/a(n) = 3*(e/7^4)^(1/7)*(Gamma(3/7) - Gamma(3/7, 1/7)). (End)

Extensions

More terms added by G. C. Greubel, Feb 23 2018

A034832 a(n) = n-th sept-factorial number divided by 5.

Original entry on oeis.org

1, 12, 228, 5928, 195624, 7824960, 367773120, 19859748480, 1211444657280, 82378236695040, 6178367752128000, 506626155674496000, 45089727855030144000, 4328613874082893824000, 445847229030538063872000, 49043195193359187025920000, 5738053837623024882032640000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A045754, A034829, A034830, A034831, A034833, A034834, A084947, A147585.

Programs

  • Mathematica
    Rest[FoldList[Times,1,7*Range[20]-2]/5] (* Harvey P. Dale, May 30 2013 *)
Drop[With[{nn = 50}, CoefficientList[Series[(-1 + (1 - 7*x)^(-5/7))/5, {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 22 2018 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace((-1 + (1-7*x)^(-5/7))/5)) \\ G. C. Greubel, Feb 22 2018

Formula

5*a(n) = (7*n-2)(!^7) = Product_{j=1..n} (7*j-2).
E.g.f.: (-1 + (1-7*x)^(-5/7))/5.
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A147585(n+1)/5.
Sum_{n>=1} 1/a(n) = 5*(e/7^2)^(1/7)*(Gamma(5/7) - Gamma(5/7, 1/7)). (End)

Extensions

More terms from G. C. Greubel, Feb 22 2018

A034831 a(n) = n-th sept-factorial number divided by 4.

Original entry on oeis.org

1, 11, 198, 4950, 158400, 6177600, 284169600, 15060988800, 903659328000, 60545174976000, 4480342948224000, 362907778806144000, 31935884534940672000, 3033909030819363840000, 309458721143575111680000, 33731000604649687173120000, 3912796070139363712081920000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A049209, A045754, A034829, A034830, A034832, A034833, A034834, A144827.

Programs

  • Magma
    [(&*[(7*k-3): k in [1..n]])/4: n in [1..30]]; // G. C. Greubel, Feb 24 2018
  • Mathematica
    Drop[With[{nn = 40}, CoefficientList[Series[(-1 + (1 - 7*x)^(-4/7))/4, {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 22 2018 *)
Table[Product[7 j - 3, {j, n}], {n, 30}]/4 (* Vincenzo Librandi, Feb 24 2018 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace((-1 + (1-7*x)^(-4/7))/4)) \\ G. C. Greubel, Feb 22 2018

Formula

4*a(n) = (7*n-3)(!^7) = Product_{j=1..n} (7*j-3).
E.g.f.: (-1 + (1-7*x)^(-4/7))/4.
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A144827(n)/4.
Sum_{n>=1} 1/a(n) = 4*(e/7^3)^(1/7)*(Gamma(4/7) - Gamma(4/7, 1/7)). (End)

Extensions

More terms added by G. C. Greubel, Feb 23 2018

A025752 7th-order Patalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 21, 637, 22295, 842751, 33429123, 1370594043, 57564949806, 2462500630590, 106872527367606, 4692675519868518, 208041948047504298, 9297874755046153626, 418404363977076913170, 18939770876029014936162, 861759574859320179595371, 39387481745040692914447251
Offset: 0

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A034833, A220607.

Programs

  • Mathematica
    CoefficientList[Series[(8 - (1 - 49*x)^(1/7))/7, {x, 0, 20}], x] (* Vincenzo Librandi, Dec 29 2012 *)
a[n_] := 49^(n-1) * Pochhammer[6/7, n-1]/n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Aug 20 2025 *)

Formula

G.f.: (8-(1-49*x)^(1/7))/7.
a(n) = 7^(n-1)*6*A034833(n-1)/n!, n >= 2, where 6*A034833(n-1)= (7*n-8)(!^7) = Product_{j=2..n} (7*j - 8). - Wolfdieter Lang
a(n) ~ 49^(n-1) / (Gamma(6/7) * n^(8/7)). - Amiram Eldar, Aug 20 2025
Showing 1-8 of 8 results.

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

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