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 1-3 of 3 results.

A086366 Number of labeled n-node digraphs in which every node belongs to a directed cycle.

Original entry on oeis.org

1, 0, 1, 18, 1699, 587940, 750744901, 3556390155318, 63740128872703879, 4405426607409460017480, 1190852520892329350092354441, 1270598627613805616203391468226138, 5381238039128882594932248239301142751179, 90766634183072089515270648224715368261615375340
Offset: 0

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Author

Keywords

Comments

These are the directed graphs whose strong components exclude a single vertex. - Andrew Howroyd, Jan 15 2022

Crossrefs

Column k=0 of A361592.
The unlabeled version is A361586.

Programs

  • PARI
    G(p)={my(n=serprec(p,x)); serconvol(p, sum(k=0, n-1, x^k/2^(k*(k-1)/2), O(x^n)))}
    U(p)={my(n=serprec(p,x)); serconvol(p, sum(k=0, n-1, x^k*2^(k*(k-1)/2), O(x^n)))}
    DigraphEgf(n)={sum(k=0, n, 2^(k*(k-1))*x^k/k!, O(x*x^n) )}
    seq(n)={Vec(serlaplace(U(1/G(exp(x+log(U(1/G(DigraphEgf(n)))))))))} \\ Andrew Howroyd, Jan 15 2022

Extensions

a(0)=1 prepended and terms a(12) and beyond from Andrew Howroyd, Jan 15 2022

A361590 Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled nodes with exactly k strongly connected components of size 1.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 5, 5, 0, 6, 90, 55, 42, 0, 31, 5289, 2451, 974, 592, 0, 302, 1071691, 323709, 94332, 29612, 15616, 0, 5984, 712342075, 135208025, 25734232, 6059018, 1650492, 795930, 0, 243668, 1585944117738, 181427072519, 21650983294, 3358042412, 704602272, 174576110, 79512478, 0, 20286025
Offset: 0

Views

Author

Andrew Howroyd, Mar 16 2023

Keywords

Examples

			Triangle begins:
        1;
        0,      1;
        1,      0,     2;
        5,      5,     0,     6;
       90,     55,    42,     0,    31;
     5289,   2451,   974,   592,     0, 302;
  1071691, 323709, 94332, 29612, 15616,   0, 5984;
  ...
		

Crossrefs

Column k=0 is A361586.
Main diagonal is A003087.
Row sums are A000273.
The labeled version is A361592.

Programs

  • PARI
    \\ See PARI link in A350794 for program code.
    { my(A=A361590triang(6)); for(n=1, #A, print(A[n])) }

A362013 Triangular array read by rows. T(n,k) is the number of labeled directed graphs on [n] with exactly k strongly connected components of size 1 with outdegree zero, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 27, 27, 9, 1, 2401, 1372, 294, 28, 1, 759375, 253125, 33750, 2250, 75, 1, 887503681, 171774906, 13852815, 595820, 14415, 186, 1, 3938980639167, 437664515463, 20841167403, 551353635, 8751645, 83349, 441, 1, 67675234241018881, 4263006881324024, 117484441611292, 1850148686792, 18210124870, 114709448, 451612, 1016, 1
Offset: 0

Views

Author

Geoffrey Critzer, Apr 03 2023

Keywords

Examples

			Triangle T(n,k) begins:
       1;
       0,      1;
       1,      2,     1;
      27,     27,     9,    1;
    2401,   1372,   294,   28,  1;
  759375, 253125, 33750, 2250, 75, 1;
  ...
		

Crossrefs

Cf. A086206 (column k=0), A053763 (row sums), A361592, A350792 (a subclass of the digraphs for the case k=1 of this sequence), A003028.

Programs

  • Mathematica
    nn = 6; B[n_] := n! 2^Binomial[n, 2] ; strong =Select[Import["https://oeis.org/A003030/b003030.txt", "Table"], Length@# == 2 &][[All, 2]]; s[z_] := Total[strong Table[z^i/i!, {i, 1, 58}]];
    ggf[egf_] := Normal[Series[egf, {z, 0, nn}]] /.Table[z^i -> z^i/2^Binomial[i, 2], {i, 0, nn}]; Table[ Take[(Table[B[n], {n, 0, nn}] CoefficientList[   Series[ggf[Exp[(u - 1) z]]/ggf[Exp[-s[z]]], {z, 0, nn}], {z, u}])[[i]], i], {i, 1, nn + 1}]
Showing 1-3 of 3 results.