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.

Previous Showing 11-18 of 18 results.

A099702 Consider the family of directed multigraphs enriched by the species of simple graphs. Sequence gives number of those multigraphs with n labeled loops and arcs.

Original entry on oeis.org

1, 2, 17, 256, 5719, 173446, 6768075, 328288840, 19468007553, 1458080017522, 183476204746761, 87209577493989776, 154656821805639801687, 617619828457724835488214, 5008102331929281541386123923, 81618549234469098721106601012472, 2666950050438611111026601803629686849
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, A014507, A098622, A099700, A099701, A099703.

Programs

  • PARI
    \\ R(n) is A006125 as e.g.f.; EnrichedGdlSeq defined in A098622.
R(n)={sum(k=0, n, 2^binomial(k, 2)*x^k/k!) + 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 A006125. - Andrew Howroyd, Jan 12 2021

Extensions

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

A099706 Consider the family of directed multigraphs enriched by the species of directed graphs. Sequence gives number of those multigraphs with n labeled loops and arcs.

Original entry on oeis.org

1, 4, 84, 3568, 305712, 87782720, 144600947392, 1139235294403328, 37012349010095737088, 4840037457225169875031040, 2535930555678883610642223895552, 5317274645187046706095607711946092544, 44602319906972740832371696997145322907873280
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, A014507, A098622, A099704, A099705, A099707.

Programs

  • PARI
    \\ R(n) is A002416 as e.g.f.; EnrichedGdlSeq defined in A098622.
R(n)={sum(k=0, n, 2^(k^2)*x^k/k!) + O(x*x^n)}
EnrichedGdlSeq(R(15)) \\ 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 A002416. - Andrew Howroyd, Jan 12 2021

Extensions

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

A099710 Consider the family of directed multigraphs enriched by the species of endofunctions. Sequence gives number of those multigraphs with n labeled loops and arcs.

Original entry on oeis.org

1, 2, 21, 372, 9503, 323528, 13976119, 740471952, 46918236113, 3486842393336, 299252510858253, 29285226572514608, 3233515108614711055, 399237909648934968160, 54699907257463871118015, 8261287679602024304387776, 1367355850924129919137226337, 246745297507913180076213875232
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, A014507, A098622, A099708, A099709, A099711.

Programs

  • PARI
    \\ R(n) is A000312 as e.g.f.; EnrichedGdlSeq defined in A098622.
R(n)={1/(1 + lambertw(-x + 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 A000312. - Andrew Howroyd, Jan 12 2021

Extensions

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

A099714 Consider the family of directed multigraphs enriched by the species of arborescences. Sequence gives number of those multigraphs with n labeled loops and arcs.

Original entry on oeis.org

1, 2, 17, 258, 5771, 174528, 6770119, 324895980, 18781627193, 1281239711000, 101465766593553, 9204346831406488, 945843113150930899, 109072242262950463552, 14001689466624210245831, 1986950788160317182000976, 309800790825415866952825137, 52786928631190620809803203872
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. A000169, A014507, A098622, A099712, A099713, A099715.

Programs

  • PARI
    \\ R(n) is A000169 as e.g.f.; EnrichedGdlSeq defined in A098622.
R(n)={-lambertw(-x + 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 R(x) is the e.g.f. of A000169. - Andrew Howroyd, Jan 12 2021

Extensions

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

A099718 Consider the family of directed multigraphs enriched by the species of trees. Sequence gives number of those multigraphs with n labeled loops and edges.

Original entry on oeis.org

1, 2, 15, 207, 4274, 120698, 4408714, 200482089, 11035845002, 719691942986, 54661283926338, 4768412660292713, 472309503983879356, 52604316569196875434, 6533611563916740388476, 898472724512273277951811, 135941600045496082012663932, 22505828691354514668620263242
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. A000272, A014507, A098622, A099716, A099717, A099719.

Programs

  • PARI
    \\ R(n) is A000272 as e.g.f.; EnrichedGdlSeq defined in A098622.
R(n)={my(w=lambertw(-x + O(x*x^n))); 1 - w - w^2/2}
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 A000272. - Andrew Howroyd, Jan 12 2021

Extensions

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

A020565 Number of cyclic oriented multigraphs on n labeled arcs (with loops).

Original entry on oeis.org

1, 2, 15, 205, 4202, 118096, 4300364, 195155304, 10727473182, 698874420944, 53040545101942, 4624423933685370, 457851029540848580, 50977215595819988320, 6329927203532081983976, 870296461701522595081624, 131659595370255359745290076
Offset: 0

Views

Author

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

Keywords

References

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

Links

Programs

  • Maple
    A020565 := proc(n)
    add((-1)^(n-k)*combinat[stirling1](n,k)*A014507(k),k=0..n) ;
end proc:
seq(A020565(n),n=0..10) ; # R. J. Mathar, Apr 30 2017
  • Mathematica
    b[n_] := Sum[StirlingS1[n, k]*BellB[2*k], {k, 0, n}];
a[n_] := Sum[(-1)^(n-k)*StirlingS1[n, k]*b[k], {k, 0, n}];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jan 21 2018, after Vladeta Jovovic *)

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*A014507(k). - Vladeta Jovovic, May 02 2004
E.g.f.: Sum(Bell(2*n)*log(1-log(1-x))^n/n!, n=0..infinity). - Vladeta Jovovic, May 02 2004
E.g.f.: exp(-1)*Sum((1-log(1-x))^(n^2)/n!,n=0..infinity). - Vladeta Jovovic, Mar 04 2008

A192666 E.g.f. satisfies: A(x) = exp(-1)*Sum_{n>=0} (1 + x*A(x))^(n^2)/n!.

Original entry on oeis.org

1, 2, 21, 444, 14415, 637268, 35822203, 2450234160, 197807272289, 18431380399184, 1948783220129813, 230702141895062720, 30251527782113610991, 4355262112839582661824, 683368350046603022039867, 116136704024677305164141056
Offset: 0

Views

Author

Paul D. Hanna, Jul 14 2011

Keywords

Comments

Compare to e.g.f. W(x) = LambertW(-x)/(-x) of A000272 (with offset) generated by: W(x) = exp(-1)*Sum_{n>=0} (1+x*W(x))^n/n! = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.

Examples

			E.g.f.: A(x) = 1 + 2*x + 21*x^2/2! + 444*x^3/3! + 14415*x^4/4! +...
where A(x) = G(x*A(x)) and A(x/G(x)) = G(x) = e.g.f. of A014507:
G(x) = 1 + 2*x + 13*x^2/2! + 162*x^3/3! + 3075*x^4/4! + 80978*x^5/5! +...

Crossrefs

Cf. A014507.

Programs

  • PARI
    /* A(x) = 1/e*Sum_{n>=0}(1+x*A(x))^(n^2)/n! (requires precision): */
{a(n)=local(A=1+x);for(i=1,n,A=exp(-1)*sum(m=0,3*n+10,(1+x*A +x*O(x^n))^(m^2)/m!));polcoeff(round(serlaplace(A+x*O(x^n))),n)}
  • PARI
    /* E.g.f. Series_Reversion(x/G(x))/x; G(x) = e.g.f. of A014507: */
{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{A014507(n)=sum(k=0, n, Stirling1(n, k)*Bell(2*k))}
{a(n)=local(G=sum(m=0,n,A014507(m)*x^m/m!)+x*O(x^n));n!*polcoeff(serreverse(x/G)/x,n)}

Formula

E.g.f.: A(x) = Series_Reversion(x/G(x))/x, where G(x) = A(x/G(x)) = e.g.f. of A014507.

A265267 E.g.f.: exp(-2) * Sum_{n>=0} 2^n * (1+x)^(n^2) / n!.

Original entry on oeis.org

1, 6, 88, 2160, 76336, 3594112, 214575872, 15695861760, 1371486918144, 140382841170944, 16572993648603136, 2228162239340027904, 337576082591565651968, 57121976918741964259328, 10713284121614206013898752, 2212342319434677836830015488, 500118162321472987555560620032, 123128345425943590420826294059008, 32864579386892803455158341264736256
Offset: 0

Views

Author

Paul D. Hanna, Jan 04 2016

Keywords

Examples

			E.g.f.: A(x) = 1 + 6*x + 88*x^2/2! + 2160*x^3/3! + 76336*x^4/4! + 3594112*x^5/5! + 214575872*x^6/6! + 15695861760*x^7/7! + 1371486918144*x^8/8! +...
where
A(x)*exp(2) = 1 + 2*(1+x) + 2^2*(1+x)^4/2! + 2^3*(1+x)^9/3! + 2^4*(1+x)^16/4! + 2^5*(1+x)^25/5! + 2^6*(1+x)^36/6! + 2^7*(1+x)^49/7! + 2^8*(1+x)^64/8! +...

Crossrefs

Cf. A014507.

Programs

  • PARI
    /* Quick print of terms 0..30: */
\p80
Vec(round( serlaplace( sum(n=0,400, 2^n * (1+x +O(x^31))^(n^2) /n! *1.)/exp(2) ) ))
Previous Showing 11-18 of 18 results.

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

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