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 1-3 of 3 results.

A317855 Decimal expansion of a constant related to the asymptotics of A122400.

Original entry on oeis.org

3, 1, 6, 1, 0, 8, 8, 6, 5, 3, 8, 6, 5, 4, 2, 8, 8, 1, 3, 8, 3, 0, 1, 7, 2, 2, 0, 2, 5, 8, 8, 1, 3, 2, 4, 9, 1, 7, 2, 6, 3, 8, 2, 7, 7, 4, 1, 8, 8, 5, 5, 6, 3, 4, 1, 6, 2, 7, 2, 7, 8, 2, 0, 7, 5, 3, 7, 6, 9, 7, 0, 5, 9, 2, 1, 9, 3, 0, 4, 6, 1, 1, 2, 1, 9, 7, 5, 7, 4, 6, 8, 5, 4, 9, 7, 8, 4, 5, 9, 3, 2, 4, 2, 2, 7
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 09 2018

Keywords

Examples

			3.161088653865428813830172202588132491726382774188556341627278...

Crossrefs

Cf. A121886, A122399, A122400, A122418, A122419, A122420, A227619, A232192, A243802, A244585, A248798, A317340, A326010.
Cf. A301584, A301585, A301586, A303056, A303057, A303654.

Programs

  • Mathematica
    r = r /. FindRoot[E^(1/r)/r + (1 + E^(1/r)) * ProductLog[-E^(-1/r)/r] == 0, {r, 3/4}, WorkingPrecision -> 120]; RealDigits[(1 + Exp[1/r])*r^2][[1]]
  • PARI
    r=solve(r=.8,1,exp(1/r)/r + (1+exp(1/r))*lambertw(-exp(-1/r)/r))
(1+exp(1/r))*r^2 \\ Charles R Greathouse IV, Jun 15 2021

Formula

Equals (1+exp(1/r))*r^2, where r = 0.873702433239668330496568304720719298213992... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0.

A122418 a(n) = Sum_{k=0..n} (k-1)^n*k!*Stirling2(n,k).

Original entry on oeis.org

1, 0, 2, 54, 2534, 186030, 19794662, 2885980734, 552803552534, 134687987183790, 40686498089484422, 14925683377452413214, 6536580413039406774134, 3368723388994026165415950, 2018248855531992511720945382, 1390953089533285777007059354494, 1092714503596231472933813958469334
Offset: 0

Views

Author

Vladeta Jovovic, Sep 03 2006

Keywords

Links

Crossrefs

Cf. A122419, A122420, A122399.

Programs

  • Maple
    A122418 := proc(n) sum((k-1)^n*k!*combinat[stirling2](n,k),k=0..n) ; end; for n from 0 to 16 do print(A122418(n)) ; od ; # R. J. Mathar, Feb 10 2007
  • Mathematica
    a[n_] := Sum[ (k-1)^n*k!*StirlingS2[n, k], {k, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Mar 26 2013 *)
  • PARI
    for(n=0,50, print1(sum(k=0,n, (k-1)^n*k!*stirling(n,k,2)), ", ")) \\ G. C. Greubel, Nov 15 2017

Formula

E.g.f.: Sum((exp((n-1)*x)-1)^n, n=0..infinity).
a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.10430562057820038909699083625848223918044424242153125547162600916636313858475... . - Vaclav Kotesovec, May 07 2014

Extensions

More terms from R. J. Mathar, Feb 10 2007

A122420 Number of labeled directed multigraphs with n arcs and with no vertex of indegree 0.

Original entry on oeis.org

1, 0, 1, 10, 120, 1778, 31685, 661940, 15882128, 430607370, 13022755068, 434697574538, 15875944361864, 629756003982336, 26963278837704185, 1239382820431888898, 60875147436141987437, 3181961834442383306068
Offset: 0

Views

Author

Vladeta Jovovic, Sep 03 2006

Keywords

Crossrefs

Cf. A104209.

Programs

  • Maple
    A122418 := proc(n) option remember ; add( combinat[stirling2](n,k)*(k-1)^n*k!,k=0..n) ; end: A122420 := proc(n) option remember ; add( abs(combinat[stirling1](n,k))*A122418(k),k=0..n)/n! ; end: for n from 0 to 30 do printf("%d, ",A122420(n)) ; od ; # R. J. Mathar, May 18 2007
  • Mathematica
    Table[1/n!*Sum[Abs[StirlingS1[n,k]]*Sum[(m-1)^k*m!*StirlingS2[k,m],{m,0,k}],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 07 2014 *)

Formula

a(n) = (1/n!)*Sum_{k=0..n} |Stirling1(n,k)|*A122418(k). G.f.: A(x/(1-x)) where A(x) is g.f. for A122419.
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.1221803955695846906452721220983425... . - Vaclav Kotesovec, May 07 2014

Extensions

More terms from R. J. Mathar, May 18 2007
Showing 1-3 of 3 results.

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