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.

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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).

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).

A091546 First column of the array A092077 ((8,2)-Stirling2).

Original entry on oeis.org

1, 56, 10192, 3872960, 2517424000, 2497284608000, 3511182158848000, 6643156644540416000, 16275733779124019200000, 50129260039701979136000000, 189588861470152885092352000000, 863766852858016544480755712000000, 4666068539139005373285042356224000000, 29489553167358513959161467691335680000000
Offset: 1

Views

Author

Wolfdieter Lang, Feb 13 2004

Keywords

Comments

Also seventh column (m=6) of triangle A091543.

Links

Crossrefs

Cf. A007559, A008542, A034689, A091543, A092077.
Cf. A073005, A175379.

Programs

  • Mathematica
    a[n_] := 6^(2*n) * Pochhammer[1/6, n] * Pochhammer[1/3, n] / 2; Array[a, 20] (* Amiram Eldar, Aug 30 2025 *)

Formula

a(n) = (2^(n-1))*Product_{j=0..n-1} ((3*j+1)*(6*j+1)), n>=1. From eq.12 of the Blasiak et al. reference with r=8, s=2, k=1.
a(n) = (6^(2*n))*risefac(1/6, n)*risefac(1/3, n)/2, n>=1, with risefac(x, n) = Pochhammer(x, n).
a(n) = fac6(6*n-5)*fac6(6*n-4)/2, n>=1, with fac6(6*n-5) = A008542(n) and fac6(6*n-4)/2 = A034689(n)= (2^(n-1))*A007559(n), (6-factorials).
a(n) ~ Pi * (6/e)^(2*n) * n^(2*n-1/2) / (Gamma(1/6) * Gamma(1/3)). - Amiram Eldar, Aug 30 2025
Previous Showing 11-13 of 13 results.

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