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 1-7 of 7 results.

A098620 Consider the family of multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled edges.

Original entry on oeis.org

1, 1, 4, 26, 257, 3586, 66207, 1540693, 43659615, 1469677309, 57681784820, 2601121752854, 133170904684965, 7664254746784243, 491679121677763607, 34905596059311761907, 2725010800987216480527, 232643959843709167832482, 21613761720729431904201734
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, A098621, A098622, A098623.

Programs

  • PARI
    \\ here R(n) is A000110 as e.g.f.
egf1(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)}
EnrichedGnSeq(R)={my(n=serprec(R, x)-1, B=exp(x/2 + O(x*x^n))*subst(egf1(n), x, log(1+x + O(x*x^n))/2)); Vec(serlaplace(subst(B, x, R-polcoef(R,0))))}
R(n)={exp(exp(x + O(x*x^n))-1)}
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 A000110. - Andrew Howroyd, Jan 12 2021

Extensions

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

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

A098627 Consider the family of directed multigraphs enriched by the species of derangements. Sequence gives number of those multigraphs with n labeled arcs.

Original entry on oeis.org

1, 0, 1, 2, 27, 164, 2335, 25458, 437241, 6965112, 145640817, 3057675290, 76814951587, 2003471245164, 59438049704943, 1855131250113498, 63937099992148785, 2327591284996635888, 91854272591000172321, 3828194864278619367474, 170846746588575658999147
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, A014505, A098623, A098624, A098625, A098626.

Programs

  • PARI
    \\ R(n) is A000166 as e.g.f.; EnrichedGdSeq defined in A098623.
R(n)={exp(-x + O(x*x^n))/(1-x)}
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 A000166. - Andrew Howroyd, Jan 12 2021

Extensions

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

A098631 Consider the family of directed multigraphs enriched by the species of parts. Sequence gives number of those multigraphs with n labeled arcs.

Original entry on oeis.org

1, 2, 28, 696, 26512, 1402656, 97017792, 8418174848, 889241719040, 111774837350912, 16420543334734848, 2778708477919836160, 535183812199464341504, 116142946557502449852416, 28156854547845767203373056, 7569375509914847295271043072, 2241898693518356603925445017600
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, A014505, A020556, A098623, A098628, A098629, A098630.

Programs

  • PARI
    \\ R(n) is A000079 as e.g.f.; EnrichedGdSeq defined in A098623.
R(n)={exp(2*x + O(x*x^n))}
EnrichedGdSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021

Formula

a(n) = 2^n*A020556(n). - Vladeta Jovovic, Aug 11 2005
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 A000079. - Andrew Howroyd, Jan 12 2021

Extensions

More terms from Vladeta Jovovic, Aug 11 2005
Terms a(14) and beyond from Andrew Howroyd, Jan 12 2021

A098639 Consider the family of directed multigraphs enriched by the species of odd sets. Sequence gives number of those multigraphs with n labeled arcs.

Original entry on oeis.org

1, 1, 6, 69, 1230, 30663, 1005692, 41571127, 2099861244, 126607647073, 8945129371976, 729628409684925, 67868881258920424, 7125522244948969319, 837004398237510194704, 109173596976047915341823, 15708090522743045757716496, 2478722722731315203268137729
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. A014505, A098623, A098636, A098637, A098638.

Programs

  • PARI
    \\ EnrichedGdSeq defined in A098623.
EnrichedGdSeq(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 A014505. - Andrew Howroyd, Jan 12 2021

Extensions

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

A099695 Consider the family of directed multigraphs enriched by the species of directed sets. Sequence gives number of those multigraphs with n labeled arcs.

Original entry on oeis.org

1, 1, 8, 106, 2144, 59844, 2173450, 98648246, 5433864078, 355229741266, 27080154837658, 2373310690810690, 236327564463489838, 26475199136060717618, 3308794737926514931894, 457980967372496137472590, 69761664006643652403884218, 11629282648335699139979015070
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. A014505, A098623, A099692, A099693, A099694.

Programs

  • PARI
    \\ R(n) is e.g.f. of 1, 1, 2, 2, 2, ...; EnrichedGdSeq defined in A098623.
R(n)={2*exp(x + O(x*x^n)) - x - 1}
EnrichedGdSeq(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 A014505. - Andrew Howroyd, Jan 12 2021

Extensions

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

A099699 Consider the family of directed multigraphs enriched by the species of involutions. Sequence gives number of those multigraphs with n labeled arcs.

Original entry on oeis.org

1, 1, 8, 108, 2200, 61708, 2249268, 102377404, 5651999688, 370171228504, 28262385542832, 2480108374814480, 247231765611893504, 27722619251007202720, 3467475213036160205984, 480277499859342401636704, 73202023124111697153718080, 12209186681659842887207280448
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, A014505, A098623, A099696, A099697, A099698.

Programs

  • PARI
    \\ R(n) is A000085 as e.g.f.; EnrichedGdSeq defined in A098623.
R(n)={exp(x+x^2/2 + O(x*x^n))}
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 A000085. - Andrew Howroyd, Jan 12 2021

Extensions

Terms a(11) and beyond from Andrew Howroyd, Jan 12 2021
Showing 1-7 of 7 results.

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

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