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 1-10 of 18 results. Next

A098622 Consider the family of directed multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled loops and arcs.

Original entry on oeis.org

1, 2, 17, 250, 5465, 162677, 6241059, 297132409, 17075153860, 1159545515804, 91501467848088, 8276847825732141, 848577193578286942, 97672164219292005480, 12518933902769241287267, 1774279753092963892540493, 276351502436571180980604240, 47046745370508674770872396843
Offset: 0

Views

Author

N. J. A. Sloane, Oct 26 2004

Keywords

References

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

Links

Crossrefs

Cf. A000110, A014507, A098620, A098621, A098623.

Programs

  • PARI
    \\ here R(n) is A000110 as e.g.f.
egfA014507(n)={my(bell=serlaplace(exp(exp(x + O(x^(2*n+1)))-1))); sum(i=0, n, sum(k=0, i, stirling(i,k,1)*polcoef(bell, 2*k))*x^i/i!) + O(x*x^n)}
EnrichedGdlSeq(R)={my(n=serprec(R, x)-1); Vec(serlaplace(subst(egfA014507(n), x, R-polcoef(R,0))))}
R(n)={exp(exp(x + O(x*x^n))-1)}
EnrichedGdlSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021

Formula

E.g.f.: exp(-1)*Sum_{n >=0} exp(n^2*(exp(x)-1))/n!. - Vladeta Jovovic, Aug 24 2006
a(n) = Sum_{k=0..n} Stirling2(n,k)*Bell(2*k). - Vladeta Jovovic, Aug 24 2006
E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014507 and 1 + R(x) is the e.g.f. of A000110. - Andrew Howroyd, Jan 12 2021

Extensions

More terms from Vladeta Jovovic, Aug 24 2006
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 15 2007
Terms a(16) and beyond from Andrew Howroyd, Jan 12 2021

A014505 Number of digraphs with unlabeled (non-isolated) nodes and n labeled edges.

Original entry on oeis.org

1, 1, 6, 68, 1206, 29982, 981476, 40515568, 2044492988, 123175320988, 8697475219688, 709097832452880, 65934837808883016, 6920436929999656936, 812724019581549433520, 105986960037601701495680
Offset: 0

Views

Author

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

Keywords

References

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

Links

Crossrefs

Cf. A014507.

Formula

E.g.f.: exp(-1) * Sum_{n>=0} (1+x)^(n^2-n) / n!. - Paul D. Hanna, Apr 25 2018
a(n) = n!*exp(-1) * Sum_{k>=sqrt(n)} binomial(k^2-k, n) / k!. - Paul D. Hanna, Apr 25 2018

A098626 Consider the family of directed multigraphs enriched by the species of derangements. Sequence gives number of those multigraphs with n labeled loops and arcs.

Original entry on oeis.org

1, 0, 2, 4, 57, 348, 5235, 57930, 1037540, 16842496, 363889755, 7792175070, 201054289293, 5345844537876, 162234861271288, 5156725529935952, 181284205622239755, 6713109719185427600, 269652617328843102055, 11418447984579685481310, 517839485352765454438270
Offset: 0

Views

Author

N. J. A. Sloane, Oct 26 2004

Keywords

References

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

Links

Crossrefs

Cf. A000166, A014507, A098622, A098624, A098625, A098627.

Programs

  • PARI
    \\ R(n) is A000166 as e.g.f.; EnrichedGdlSeq defined in A098622.
R(n)={exp(-x + O(x*x^n))/(1-x)}
EnrichedGdlSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021

Formula

E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014507 and 1 + R(x) is the e.g.f. of A000166. - Andrew Howroyd, Jan 12 2021

Extensions

Terms a(14) and beyond from Andrew Howroyd, Jan 12 2021

A098630 Consider the family of directed multigraphs enriched by the species of parts. Sequence gives number of those multigraphs with n labeled loops and arcs.

Original entry on oeis.org

1, 4, 60, 1624, 66240, 3711200, 269670208, 24435113216, 2682916389632, 349223324753408, 52965538033020928, 9229753832340117504, 1826647528631522463744, 406579171521484851396608, 100934277604965329345822720, 27746271707522968205726416896
Offset: 0

Views

Author

N. J. A. Sloane, Oct 26 2004

Keywords

References

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

Links

Crossrefs

Cf. A000079, A014507, A098622, A098628, A098629, A098631.

Programs

  • PARI
    a(n) = {2^n*sum(k=0, 2*n, stirling(2*n,k,2))} \\ Andrew Howroyd, Jan 12 2021
  • PARI
    \\ R(n) is A000079 as e.g.f.; EnrichedGdlSeq defined in A098622.
R(n)={exp(2*x + O(x*x^n))}
EnrichedGdlSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021

Formula

a(n) = 2^n*Bell(2*n). - Vladeta Jovovic, Aug 22 2006
E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014507 and 1 + R(x) is the e.g.f. of A000079. - Andrew Howroyd, Jan 12 2021

Extensions

Terms a(11) and beyond from Andrew Howroyd, Jan 12 2021

A098638 Consider the family of directed multigraphs enriched by the species of odd sets. Sequence gives number of those multigraphs with n labeled loops and arcs.

Original entry on oeis.org

1, 2, 13, 164, 3127, 82600, 2845775, 122820136, 6446913953, 402413160952, 29343933156485, 2464029760993520, 235446319553848087, 25346231173047308256, 3047931031445529965527, 406412844141860523543392, 59704680455100785101683457, 9608818815170839730520275488
Offset: 0

Views

Author

N. J. A. Sloane, Oct 26 2004

Keywords

References

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

Links

Crossrefs

Cf. A014507, A098622, A098636, A098637, A098639.

Programs

  • PARI
    \\ EnrichedGdlSeq defined in A098622.
EnrichedGdlSeq(sinh(x + O(x*x^20))) \\ Andrew Howroyd, Jan 12 2021

Formula

E.g.f.: exp(-1)*Sum_{n>=0}(1+sinh(x))^(n^2)/n!. - Vladeta Jovovic, Mar 04 2008
E.g.f.: B(sinh(x)) where B(x) is the e.g.f. of A014507. - Andrew Howroyd, Jan 12 2021

Extensions

Missing a(10) inserted and terms a(13) and beyond from Andrew Howroyd, Jan 12 2021

A099694 Consider the family of directed multigraphs enriched by the species of directed sets. Sequence gives number of those multigraphs with n labeled loops and arcs.

Original entry on oeis.org

1, 2, 17, 244, 5283, 156092, 5954547, 282221828, 16159327961, 1094056231572, 86116276633357, 7773114989571400, 795480206815177651, 91417037615848058160, 11701283925663217478843, 1656436690705751478232180, 257730676653629520748175377, 43837005194184348815823808500
Offset: 0

Views

Author

N. J. A. Sloane, Oct 26 2004

Keywords

References

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

Links

Crossrefs

Cf. A014507, A098622, A099692, A099693, A099695.

Programs

  • PARI
    \\ R(n) is e.g.f. of 1, 1, 2, 2, 2, ...; EnrichedGdlSeq defined in A098622.
R(n)={2*exp(x + O(x*x^n)) - x - 1}
EnrichedGdlSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021

Formula

E.g.f.: B(2*exp(x) - x - 2) where B(x) is the e.g.f. of A014507. - Andrew Howroyd, Jan 12 2021

Extensions

Terms a(11) and beyond from Andrew Howroyd, Jan 12 2021

A099698 Consider the family of directed multigraphs enriched by the species of involutions. Sequence gives number of those multigraphs with n labeled loops and arcs.

Original entry on oeis.org

1, 2, 17, 248, 5403, 160420, 6142567, 291996934, 16759322733, 1136940595762, 89641455771637, 8102778995663368, 830222723124364047, 95509354134959796556, 12236166882713532940611, 1733521075683722202738222, 269910543278748394820341769, 45936441912756036235229989058
Offset: 0

Views

Author

N. J. A. Sloane, Jun 25 2017

Keywords

Links

Crossrefs

Cf. A000085, A014507, A098622, A099696, A099697, A099699.

Programs

  • PARI
    \\ R(n) is A000085 as e.g.f.; EnrichedGdlSeq defined in A098622.
R(n)={exp(x+x^2/2 + O(x*x^n))}
EnrichedGdlSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021

Formula

E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014507 and 1 + R(x) is the e.g.f. of A000085. - Andrew Howroyd, Jan 12 2021

Extensions

Dead sequence restored, corrected and extended by Andrew Howroyd, Jan 12 2021

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!.

A211252 E.g.f.: exp(-1)*Sum_{n>=0} (1+x)^(n^5)/n!.

Original entry on oeis.org

1, 52, 115923, 1382610724, 51715861759515, 4638073139045397206, 846679440053068198564757, 281582422101970811697025996458, 157442703858164474987714673019721909, 139252837198831456324098952617013102583100, 185718002275320639405130518085966960592675564591
Offset: 0

Views

Author

Paul D. Hanna, Apr 07 2012

Keywords

Examples

			E.g.f.: A(x) = 1 + 52*x + 115923*x^2/2! + 1382610724*x^3/3! + 51715861759515*x^4/4! +...
such that
A(x) = exp(-1)*(1 + (1+x) + (1+x)^32/2! + (1+x)^243/3! + (1+x)^1024/4! +...).

Crossrefs

Cf. A000110 (Bell), A014507, A211250, A211251.

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(5*k))}
for(n=0,15,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} Stirling1(n, k)*Bell(5*k).
a(n) = n!*exp(-1)*Sum_{k>=[n^(1/5)]} binomial(k^5,n)/k!.
Showing 1-10 of 18 results. Next

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

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