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 1-6 of 6 results.

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

A034833 a(n) = n-th sept-factorial number divided by 6.

Original entry on oeis.org

1, 13, 260, 7020, 238680, 9785880, 469722240, 25834723200, 1601752838400, 110520945849600, 8399591884569600, 697166126419276800, 62744951377734912000, 6086260283640286464000, 632971069498589792256000, 70259788714343466940416000, 8290655068292529098969088000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A049209, A045754, A034829, A034830, A034831, A034832, A034834.

Programs

  • Magma
    [(&*[(7*k-1): k in [1..n]])/6: n in [1..30]]; // G. C. Greubel, Feb 24 2018
  • Mathematica
    FoldList[Times,1,Rest[7*Range[20]-1]] (* Harvey P. Dale, Dec 15 2014 *)
  • PARI
    my(x='x+('x^30)); Vec(serlaplace((-1 + (1-7*x)^(-6/7))/6)) \\ G. C. Greubel, Feb 22 2018

Formula

a(n) = A049209(n)/6;
a(n) = (7*n-1)(!^7)/6;
a(n) = (1/6)*Product_{j=1..n} (7*j-1);
a(n) = (1/6)*(7*n)! / (7^n * n! * A045754(n) * 2*A034829(n) * 3*A034830(n) * 4*A034831(n) * 5*A034832(n)).
E.g.f.: (-1 + (1-7*x)^(-6/7))/6.
Sum_{n>=1} 1/a(n) = 6*(e/7)^(1/7)*(Gamma(6/7) - Gamma(6/7, 1/7)). - Amiram Eldar, Dec 20 2022

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
Showing 1-6 of 6 results.

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