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 1-4 of 4 results.

A054941 Number of weakly connected oriented graphs on n labeled nodes.

Original entry on oeis.org

1, 2, 20, 624, 55248, 13982208, 10358360640, 22792648882176, 149888345786341632, 2952810709943411146752, 174416705255313941476193280, 30901060796613886817249881227264, 16422801513633911416125344647746244608, 26183660776604240464418800095675915958222848
Offset: 1

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Author

N. J. A. Sloane, May 24 2000

Keywords

Comments

The triangle of oriented labeled graphs on n>=1 nodes with 1<=k<=n components and row sums A047656 starts:
1;
2, 1;
20, 6, 1;
624, 92, 12, 1;
55248, 3520, 260, 20, 1;
13982208, 354208, 11880, 580, 30, 1; - R. J. Mathar, Apr 29 2019

Links

Crossrefs

Row sums of A350732.
The unlabeled version is A086345.
Cf. A001187 (graphs), A003027 (digraphs), A350730 (strongly connected).

Programs

  • Magma
    m:=30;
f:= func< x | (&+[3^Binomial(n,2)*x^n/Factorial(n) : n in [0..m+3]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( Log(f(x)) ))); // G. C. Greubel, Apr 28 2023
  • Mathematica
    nn=20; s=Sum[3^Binomial[n,2]x^n/n!,{n,0,nn}];
Drop[Range[0,nn]! CoefficientList[Series[Log[s]+1,{x,0,nn}],x],1] (* Geoffrey Critzer, Oct 22 2012 *)
  • PARI
    N=20; x='x+O('x^N); Vec(serlaplace(log(sum(k=0, N, 3^binomial(k, 2)*x^k/k!)))) \\ Seiichi Manyama, May 18 2019
  • SageMath
    m=30
def f(x): return sum(3^binomial(n,2)*x^n/factorial(n) for n in range(m+4))
def A054941_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( log(f(x)) ).egf_to_ogf().list()
a=A054941_list(40); a[1:] # G. C. Greubel, Apr 28 2023

Formula

E.g.f.: log( Sum_{n >= 0} 3^binomial(n, 2)*x^n/n! ). - Vladeta Jovovic, Feb 14 2003

Extensions

More terms from Vladeta Jovovic, Feb 14 2003

A350734 Triangle read by rows: T(n,k) is the number of weakly connected oriented graphs on n unlabeled nodes with k arcs, n >= 1, k = 0..n*(n-1)/2.

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 2, 0, 0, 0, 8, 12, 10, 4, 0, 0, 0, 0, 27, 68, 127, 144, 107, 50, 12, 0, 0, 0, 0, 0, 91, 395, 1144, 2393, 3767, 4500, 4112, 2740, 1274, 376, 56, 0, 0, 0, 0, 0, 0, 350, 2170, 9139, 28606, 71583, 145600, 244589, 339090, 387458, 361394, 271177, 159872, 71320, 22690, 4604, 456
Offset: 1

Views

Author

Andrew Howroyd, Jan 13 2022

Keywords

Examples

			Triangle begins:
  [1] 1;
  [2] 0, 1;
  [3] 0, 0, 3, 2;
  [4] 0, 0, 0, 8, 12, 10,   4;
  [5] 0, 0, 0, 0, 27, 68, 127, 144, 107, 50, 12;
  ...

Links

Crossrefs

Row sums are A086345.
Column sums are A350915.
Leading diagonal is A000238.
The labeled version is A350732.
Cf. A054733, A350733, A350750, A350914 (transpose).

Programs

  • PARI
    InvEulerMTS(p)={my(n=serprec(p,x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i)}
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, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2))}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+2*x^i)); s/n!}
row(n)={Vecrev(polcoef(InvEulerMTS(sum(i=0, n, G(i, y)*x^i, O(x*x^n))), n))}
{ for(n=1, 6, print(row(n))) }

A350731 Triangle read by rows: T(n,k) is the number of strongly connected oriented graphs on n labeled nodes with k arcs, n >= 1, k=0..n*(n-1)/2.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 6, 36, 24, 0, 0, 0, 0, 0, 24, 480, 1940, 2970, 2040, 544, 0, 0, 0, 0, 0, 0, 120, 5040, 51330, 221910, 527940, 772080, 722250, 426420, 146160, 22320, 0, 0, 0, 0, 0, 0, 0, 720, 52920, 1026060, 8810970, 43268442, 138984510
Offset: 1

Views

Author

Andrew Howroyd, Jan 11 2022

Keywords

Examples

			Triangle begins:
  [1] 1;
  [2] 0, 0;
  [3] 0, 0, 0, 2;
  [4] 0, 0, 0, 0, 6, 36,  24;
  [5] 0, 0, 0, 0, 0, 24, 480, 1940, 2970, 2040, 544;
  ...

Links

Crossrefs

Row sums are A350730.
The unlabeled version is A350750.
Cf. A057273 (digraphs), A350732 (weakly connected).

Programs

  • PARI
    OrientedGgf(n, y=1) = {sum(k=0, n, ((1+2*y)/(1+y))^(k*(k-1)/2)*x^k/k!, O(x*x^n) )}
StrongO(n, y=1) = {my(g=serconvol(1/OrientedGgf(n,y), sum(k=0, n, x^k*(1+y)^(k*(k-1)/2), O(x*x^n)))); Vec(serlaplace(-log(g)))}
row(n)={Vecrev(StrongO(n,'y)[n], n*(n-1)/2+1)}
{ for(n=1, 6, print(row(n))) }

A350749 Triangle read by rows: T(n,k) is the number of oriented graphs on n labeled nodes with k arcs, n >= 0, k = 0..n*(n-1)/2.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 12, 8, 1, 12, 60, 160, 240, 192, 64, 1, 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024, 1, 30, 420, 3640, 21840, 96096, 320320, 823680, 1647360, 2562560, 3075072, 2795520, 1863680, 860160, 245760, 32768
Offset: 0

Views

Author

Andrew Howroyd, Feb 15 2022

Keywords

Examples

			Triangle begins:
  [0] 1;
  [1] 1;
  [2] 1,  2;
  [3] 1,  6,  12,   8;
  [4] 1, 12,  60, 160,  240,  192,    64;
  [5] 1, 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024;
  ...

Links

Crossrefs

Row sums are A047656.
The unlabeled version is A350733.
Cf. A013609, A350732 (weakly connected), A350731 (strongly connected).

Programs

  • PARI
    T(n,k) = 2^k * binomial(n*(n-1)/2, k)
  • PARI
    row(n) = {Vecrev((1+2*y)^(n*(n-1)/2))}
{ for(n=0, 6, print(row(n))) }

Formula

T(n,k) = 2^k * binomial(n*(n-1)/2, k) = A013609(n*(n-1)/2, k).
T(n,k) = [y^k] (1+2*y)^(n*(n-1)/2).
Showing 1-4 of 4 results.

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

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