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.

A014507 Number of digraphs with loops, having unlabeled (non-isolated) nodes and n labeled edges.

Original entry on oeis.org

1, 2, 13, 162, 3075, 80978, 2784067, 119971162, 6289972169, 392257225754, 28582571639293, 2398695602082442, 229094801646110203, 24652935339990534970, 2963620352166634246995, 395067805289398293647026, 58025593661340099613984593, 9336949406574071339557552946
Offset: 0

Views

Author

Simon Plouffe, Gilbert Labelle (gilbert(AT)lacim.uqam.ca)

Keywords

References

  • G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Links

Crossrefs

Cf. A000110 (Bell), A211250, A211251, A211252.

Programs

  • Maple
    A014507 := proc(n)
    add(combinat[stirling1](n,k)*combinat[bell](2*k),k=0..n) ;
end proc:
seq(A014507(n),n=0..10) ; # R. J. Mathar, Apr 30 2017
  • Mathematica
    a[n_] := Sum[StirlingS1[n, k]*BellB[2*k], {k, 0, n}];
Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jan 21 2018, from Vladeta Jovovic's formula *)
  • PARI
    /* From Vladeta Jovovic's formula: */
{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1),n)}
{a(n)=sum(k=0, n, Stirling1(n, k)*Bell(2*k))}

Formula

a(n) = Sum_{k=0..n} Stirling1(n, k)*Bell(2*k). - Vladeta Jovovic, Jun 21 2003
E.g.f.: exp(-1)*Sum_{n>=0} (1+x)^(n^2)/n!. - Paul D. Hanna, Jul 03 2011
a(n) = n!*exp(-1)*Sum_{k>=sqrt(n)} binomial(k^2,n)/k!. - Paul D. Hanna, Jul 03 2011

A211250 E.g.f.: exp(-1)*Sum_{n>=0} (1+x)^(n^3)/n!.

Original entry on oeis.org

1, 5, 198, 20548, 4088918, 1341552690, 661685880676, 460785157967228, 432879460822014552, 529918744425680488240, 822575286838815581568992, 1583737023708711008926884072, 3713773722396456674797120593784, 10445266376618187161982580673417192
Offset: 0

Views

Author

Paul D. Hanna, Apr 07 2012

Keywords

Examples

			E.g.f.: A(x) = 1 + 5*x + 198*x^2/2! + 20548*x^3/3! + 4088918*x^4/4! +...
such that
A(x) = exp(-1)*(1 + (1+x) + (1+x)^8/2! + (1+x)^27/3! + (1+x)^64/4! +...).

Crossrefs

Cf. A000110 (Bell), A014507, A211251, A211252.

Programs

  • PARI
    {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{a(n)=sum(k=0, n, Stirling1(n, k)*Bell(3*k))}
for(n=0,15,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} Stirling1(n, k)*Bell(3*k).
a(n) = n!*exp(-1)*Sum_{k>=[n^(1/3)]} binomial(k^3,n)/k!.

A211251 E.g.f.: exp(-1)*Sum_{n>=0} (1+x)^(n^4)/n!.

Original entry on oeis.org

1, 15, 4125, 4201207, 10454906015, 51619504083157, 445183896786430439, 6151183312376366042809, 127892318444027363237894001, 3815107763405827557700743314007, 157278812586433713743644391748289829, 8693308684725580082237757157480179540583
Offset: 0

Views

Author

Paul D. Hanna, Apr 07 2012

Keywords

Examples

			E.g.f.: A(x) = 1 + 15*x + 4125*x^2/2! + 4201207*x^3/3! + 10454906015*x^4/4! +...
such that
A(x) = exp(-1)*(1 + (1+x) + (1+x)^16/2! + (1+x)^81/3! + (1+x)^256/4! +...).

Crossrefs

Cf. A000110 (Bell), A014507, A211250, A211252.

Programs

  • PARI
    {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{a(n)=sum(k=0, n, Stirling1(n, k)*Bell(4*k))}
for(n=0,15,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} Stirling1(n, k)*Bell(4*k).
a(n) = n!*exp(-1)*Sum_{k>=[n^(1/4)]} binomial(k^4,n)/k!.
Showing 1-3 of 3 results.

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