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 12 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

A098623 Consider the family of directed multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled arcs.

Original entry on oeis.org

1, 1, 8, 109, 2229, 62684, 2289151, 104344153, 5767234550, 378073098155, 28888082263581, 2536660090249102, 253007765488793325, 28383529110762969901, 3551558435250676339536, 492092920443604792460905, 75025155137863150912784409, 12516480979952118669729618300
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, A098620, A098621, A098622.

Programs

  • PARI
    \\ here R(n) is A000110 as e.g.f.
egfA020556(n)={my(bell=serlaplace(exp(exp(x + O(x^(2*n+1)))-1))); sum(i=0, n, sum(k=0, i, (-1)^k*binomial(i, k)*polcoef(bell, 2*i-k))*x^i/i!) + O(x*x^n)}
EnrichedGdSeq(R)={my(n=serprec(R, x)-1, B=subst(egfA020556(n), x, log(1+x + O(x*x^n)))); Vec(serlaplace(subst(B, x, R-polcoef(R,0))))}
R(n)={exp(exp(x + O(x*x^n))-1)}
EnrichedGdSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021

Formula

E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014505 and 1 + R(x) is the e.g.f. of A000110. - Andrew Howroyd, Jan 12 2021

Extensions

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

A098624 Consider the family of multigraphs enriched by the species of derangements. Sequence gives number of those multigraphs with n labeled edges.

Original entry on oeis.org

1, 0, 1, 2, 15, 84, 750, 6852, 79639, 1006184, 14875218, 241078100, 4392257716, 87279581232, 1905609327583, 45008114794874, 1150897256534370, 31580332783936416, 928535967078634497, 29090873853321687666, 969132936087009709174, 34198721664081728281400
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, A014500, A098620, A098625, A098626, A098627.

Programs

  • PARI
    \\ R(n) is A000166 as e.g.f.; EnrichedGnSeq defined in A098620.
R(n)={exp(-x + O(x*x^n))/(1-x)}
EnrichedGnSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021

Formula

E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014500 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

A098628 Consider the family of multigraphs enriched by the species of parts. Sequence gives number of those multigraphs with n labeled edges.

Original entry on oeis.org

1, 2, 12, 128, 2224, 56000, 1880832, 79985792, 4161468928, 258415579648, 18793653411840, 1576791247634432, 150745211441983488, 16253127712884269056, 1959064946185017851904, 262002352633857351942144, 38624060984664180255621120, 6240304185636529522872025088
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, A014500, A098620, A098629, A098630, A098631.

Programs

  • PARI
    \\ R(n) is A000079 as e.g.f.; EnrichedGnSeq defined in A098620.
R(n)={exp(2*x + O(x*x^n))}
EnrichedGnSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021

Formula

E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014500 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

A098636 Consider the family of multigraphs enriched by the species of odd sets. Sequence gives number of those multigraphs with n labeled edges.

Original entry on oeis.org

1, 1, 2, 10, 78, 885, 13487, 261848, 6255453, 179297990, 6046396379, 236175330388, 10549286540957, 533103416306743, 30203144498636380, 1903491404510540902, 132543022174482851436, 10136316177816553484295, 846893706267135762556915, 76941424170126460702604994
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. A014500, A098620, A098637, A098638, A098639.

Programs

  • PARI
    \\ EnrichedGnSeq defined in A098620.
EnrichedGnSeq(sinh(x + O(x*x^20))) \\ Andrew Howroyd, Jan 12 2021

Formula

E.g.f.: B(sinh(x)) where B(x) is the e.g.f. of A014500. - Andrew Howroyd, Jan 12 2021

Extensions

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

A099692 Consider the family of multigraphs enriched by the species of directed sets. Sequence gives number of those multigraphs with n labeled edges.

Original entry on oeis.org

1, 1, 4, 23, 220, 3016, 55011, 1265824, 35496711, 1183686987, 46072834777, 2062557088117, 104926356851165, 6004962409831577, 383331023991407286, 27094756978689827593, 2107021273883402908850, 179261681391054814324774, 16602830645109035036038335
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, A014500, A098620, A099693, A099694, A099695.

Programs

  • PARI
    \\ R(n) is e.g.f. of 1,1,2,2,2,2,...; EnrichedGnSeq defined in A098620.
R(n)={2*exp(x + O(x*x^n)) - x - 1}
EnrichedGnSeq(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 A014500. - Andrew Howroyd, Jan 12 2021

Extensions

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

A099696 Consider the family of multigraphs enriched by the species of involutions. Sequence gives number of those multigraphs with n labeled edges.

Original entry on oeis.org

1, 1, 4, 25, 244, 3380, 62133, 1440382, 40673705, 1364815169, 53415511305, 2402797049419, 122751622204827, 7051227704802797, 451598420376965588, 32013004761567761223, 2495936511077175475140, 212840593118800653411004, 19753575434503894710824531
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. A000085, A014500, A098620, A099697, A099698, A099699.

Programs

  • PARI
    \\ R(n) is A000085 as e.g.f.; EnrichedGnSeq defined in A098620.
R(n)={exp(x+x^2/2 + O(x*x^n))}
EnrichedGnSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021

Formula

E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014500 and 1 + R(x) is the e.g.f. of A000085. - Andrew Howroyd, Jan 12 2021

Extensions

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

A099700 Consider the family of multigraphs enriched by the species of simple graphs. Sequence gives number of those multigraphs with n labeled edges.

Original entry on oeis.org

1, 1, 4, 29, 330, 5438, 128211, 4808964, 378829853, 77137284917, 36854103598061, 36864364745783295, 74684573193253556537, 304187997559381840229969, 2484431769481244742219110666, 40639512967159110848931023115111, 1330529956364528398902155692019721596
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. A006125, A014500, A098620, A099701, A099702, A099703.

Programs

  • PARI
    \\ R(n) is A006125 as e.g.f.; EnrichedGnSeq defined in A098620.
R(n)={sum(k=0, n, 2^binomial(k,2)*x^k/k!) + O(x*x^n)}
EnrichedGnSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021

Formula

E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014500 and 1 + R(x) is the e.g.f. of A006125. - Andrew Howroyd, Jan 12 2021

Extensions

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

A099704 Consider the family of multigraphs enriched by the species of directed graphs. Sequence gives number of those multigraphs with n labeled edges.

Original entry on oeis.org

1, 2, 24, 776, 79840, 35397440, 69619053504, 564929183555840, 18464894708236907776, 2418517115222622481308160, 1267747370909677813160722947072, 2658511777246500251150215101758228480, 22300872810108738542496498718468714032205824
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. A002416, A014500, A098620, A099705, A099706, A099707.

Programs

  • PARI
    \\ R(n) is A002416 as e.g.f.; EnrichedGnSeq defined in A098620.
R(n)={sum(k=0, n, 2^(k^2)*x^k/k!) + O(x*x^n)}
EnrichedGnSeq(R(15)) \\ Andrew Howroyd, Jan 12 2021

Formula

E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014500 and 1 + R(x) is the e.g.f. of A002416. - Andrew Howroyd, Jan 12 2021

Extensions

Terms a(10) and beyond from Andrew Howroyd, Jan 12 2021

A099708 Consider the family of multigraphs enriched by the species of endofunctions. Sequence gives number of those multigraphs with n labeled edges.

Original entry on oeis.org

1, 1, 6, 60, 854, 16029, 378871, 10926690, 375538541, 15097900582, 699359781567, 36859422340308, 2187121403805853, 144804645827958839, 10615679263174481480, 856040905847508506792, 75495130803739278866508, 7244702305184037575057831, 753093536414613689614872227
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. A000312, A014500, A098620, A099709, A099710, A099711.

Programs

  • PARI
    \\ R(n) is A000312 as e.g.f.; EnrichedGnSeq defined in A098620.
R(n)={1/(1 + lambertw(-x + O(x*x^n)))}
EnrichedGnSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021

Formula

E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014500 and 1 + R(x) is the e.g.f. of A000312. - Andrew Howroyd, Jan 12 2021

Extensions

Terms a(12) and beyond from Andrew Howroyd, Jan 12 2021
Showing 1-10 of 12 results. Next

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

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