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 11-15 of 15 results.

A101389 Number of n-vertex unlabeled oriented graphs without endpoints.

Original entry on oeis.org

1, 1, 1, 3, 21, 369, 16929, 1913682, 546626268, 406959998851, 808598348346150, 4358157210587930509, 64443771774627635711718, 2636248889492487709302815665, 300297332862557660078111708007894, 95764277032243987785712142452776403618, 85885545190811866954428990373255822969983915
Offset: 0

Views

Author

Goran Kilibarda, Vladeta Jovovic, Jan 14 2005

Keywords

Examples

			a(3) = 3 because there are 2 distinct orientations of the triangle K_3 plus the empty graph on 3 vertices.

Links

Crossrefs

Cf. A100569 (labeled case), A100548, A101388, A004110, A004108, A059166, A059167, A001174.

Programs

  • PARI
    \\ See links in A339645 for combinatorial species functions.
oedges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, (v[i]-1)\2)}
ographsCycleIndex(n)={my(s=0); forpart(p=n, s+=permcount(p) * 3^oedges(p) * sMonomial(p)); s/n!}
ographs(n)={sum(k=0, n, ographsCycleIndex(k)*x^k) + O(x*x^n)}
trees(n,k)={sRevert(x*sv(1)/sExp(k*x*sv(1) + O(x^n)))}
cycleIndexSeries(n)={my(g=ographs(n), tr=trees(n,2), tu=tr-tr^2); sSolve( g/sExp(tu), tr )*symGroupSeries(n)}
NumUnlabeledObjsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 27 2020
  • PARI
    \\ faster stand-alone version
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, (v[i]-1)\2)}
seq(n)={Vec(sum(k=0, n, my(s=0); forpart(p=k, s+=permcount(p) * 3^edges(p) * prod(i=1, #p, my(d=p[i]); (1-x^d)^2 + O(x*x^(n-k))) ); x^k*s/k!)/(1-x^2))} \\ Andrew Howroyd, Jan 22 2021

Extensions

a(0)=1 prepended and terms a(9) and beyond from Andrew Howroyd, Dec 27 2020

A101460 Number of connected antisymmetric relations on n unlabeled nodes.

Original entry on oeis.org

1, 2, 4, 32, 467, 15726, 1283648, 266995482, 145658273814, 212067643326874, 834200554714504905, 8952922576975709358534, 264287923205519914989471726, 21606715274151098406493353524694, 4921011141817073607674347572008576367
Offset: 0

Views

Author

Vladeta Jovovic, Jan 20 2005

Keywords

Links

Crossrefs

Cf. A083670, A086345, A001174.

Formula

Inverse Euler transform of A083670. - Andrew Howroyd, Oct 24 2019

A281446 Triangle read by rows: Number of oriented graphs on n nodes with k components.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 1, 1, 0, 34, 6, 1, 1, 0, 535, 39, 6, 1, 1, 0, 20848, 584, 40, 6, 1, 1, 0, 2120098, 21553, 589, 40, 6, 1, 1, 0, 572849763, 2144216, 21602, 590, 40, 6, 1, 1, 0, 415361983540, 575092291, 2144956, 21607, 590, 40, 6, 1, 1, 0, 815590925440865, 415946286005, 575116919, 2145005, 21608
Offset: 0

Views

Author

R. J. Mathar, Apr 13 2017

Keywords

Comments

Multiset transform of A086345.

Examples

			1;
0,1;
0,1,1;
0,5,1,1;
0,34,6,1,1;
0,535,39,6,1,1;
0,20848,584,40,6,1,1;
0,2120098,21553,589,40,6,1,1;
0,572849763,2144216,21602,590,40,6,1,1;
0,415361983540,575092291,2144956,21607,590,40,6,1,1;

Links

Crossrefs

Cf. A086345 (column 1), A001174 (row sums).

Formula

G.f.: Product_{j>=1} (1-y*x^j)^(-A086345(j)). - Alois P. Heinz, Apr 13 2017

A054934 Number of oriented graphs on n nodes up to reversing the arcs.

Original entry on oeis.org

1, 2, 6, 30, 342, 11164, 1077370, 287640989, 207974848520, 408004023529326, 2187203136795598146, 32269918474347692838600, 1318898367787065334889143452, 150182948079490899321955309512894, 47886343174490577986560743878301096450, 42944209124580582731273744197913175367709988
Offset: 1

Views

Author

N. J. A. Sloane, May 24 2000

Keywords

Links

Crossrefs

Cf. A001174, A005639.

Programs

Formula

Average of A001174 and A005639.

Extensions

More terms from Philip Sung (phil(AT)main.nu), May 07 2001
Terms a(15) and beyond from Andrew Howroyd, Sep 16 2018

A126121 Numerators of sequence of fractions with e.g.f. sqrt(1+x)/(1-x)^2.

Original entry on oeis.org

1, 5, 31, 255, 2577, 31245, 439695, 7072695, 127699425, 2562270165, 56484554175, 1358576240175, 35374065613425, 992016072172125, 29792674421484975, 954480422711190375, 32479589325536978625, 1170329273010701929125, 44502357662442514209375, 1781390379962467540641375
Offset: 0

Views

Author

N. J. A. Sloane, Mar 22 2007

Keywords

Comments

Denominators are successive powers of 2.

Examples

			The fractions are 1, 5/2, 31/4, 255/8, 2577/16, 31245/32, 439695/64, ...

Links

Crossrefs

Cf. A003148, A001174, A126115, A126119.

Programs

  • Mathematica
    With[{nn=20},Numerator[CoefficientList[Series[Sqrt[1+x]/(1-x)^2,{x, 0, nn}], x] Range[0,nn]!]] (* Harvey P. Dale, Jan 29 2016 *)
  • PARI
    x='x+O('x^25); Vec(serlaplace(sqrt(1+2*x)/(1-2*x)^2)) \\ G. C. Greubel, May 25 2017

Formula

E.g.f.: 1/G(0) where G(k) = 1 - 4*x/(1 + x/(1 - x - (2*k+1)/( 2*k+1 - 4*(k+1)*x/G(k+1)))); (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Jul 28 2012
From Benedict W. J. Irwin, May 19 2016: (Start)
E.g.f.: sqrt(1+2*x)/(1-2*x)^2.
a(n) = (-1)^(n+1)*2^(n-1)*(n-3/2)!*2F1(2,-n;(3/2)-n;-1)/sqrt(Pi).
(End)
D-finite with recurrence a(n) -5*a(n-1) -2*(2*n-1)*(n-1)*a(n-2)=0. - R. J. Mathar, Feb 08 2021
Previous Showing 11-15 of 15 results.

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

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