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 1-5 of 5 results.

A002031 Number of labeled connected digraphs on n nodes where every node has indegree 0 or outdegree 0 and no isolated nodes.

Original entry on oeis.org

2, 6, 38, 390, 6062, 134526, 4172198, 178449270, 10508108222, 853219059726, 95965963939958, 15015789392011590, 3282145108526132942, 1005193051984479922206, 432437051675617901246918, 261774334771663762228012950, 223306437526333657726283273822
Offset: 2

Views

Author

N. J. A. Sloane

Keywords

Comments

Also number of labeled connected graphs with 2-colored nodes with no isolated nodes where black nodes are only connected to white nodes and vice versa.
In- or outdegree zero implies loops are not admitted. Multi-arcs are not admitted. - R. J. Mathar, Nov 18 2023

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001831, A001832, A002032, A047863, A052332, A007776 (unlabeled case). Essentially the same as A002027.

Programs

  • Maple
    logtr:= proc(p) local b; b:=proc(n) option remember; local k; if n=0 then 1 else p(n)- add(k *binomial(n,k) *p(n-k) *b(k), k=1..n-1)/n fi end end: digr:= n-> add(binomial(n,k) *(2^k-2)^(n-k), k=0..n): a:= logtr(digr): seq(a(n), n=2..25);  # Alois P. Heinz, Sep 14 2008
  • Mathematica
    terms = 17; s = Log[Sum[Exp[(2^n - 2)*x]*(x^n/n!), {n, 0, terms+2}]] + O[x]^(terms+2); Drop[CoefficientList[s, x]*Range[0, terms+1]!, 2] (* Jean-François Alcover, Nov 08 2011, after Vladeta Jovovic, updated Jan 12 2018 *)

Formula

Logarithmic transform of A052332.
E.g.f.: log(Sum(exp((2^n-2)*x)*x^n/n!, n=0..infinity)). - Vladeta Jovovic, May 28 2004
a(n) = f(n,2) using functions defined in A002032. - Sean A. Irvine, May 29 2013

Extensions

More terms, formula and new title from Christian G. Bower, Dec 15 1999
Corrected by Vladeta Jovovic, Apr 12 2003

A002027 Number of connected graphs on n labeled nodes, each node being colored with one of 2 colors, such that no edge joins nodes of the same color.

Original entry on oeis.org

1, 2, 2, 6, 38, 390, 6062, 134526, 4172198, 178449270, 10508108222, 853219059726, 95965963939958, 15015789392011590, 3282145108526132942, 1005193051984479922206, 432437051675617901246918, 261774334771663762228012950, 223306437526333657726283273822
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) is the number of connected labeled graphs with n 2-colored nodes where black nodes are only connected to white nodes and vice versa. - Geoffrey Critzer, Sep 05 2013

References

  • R. C. Read, personal communication.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column k=2 of A322279.
Essentially the same as A002031.
Cf. A002032.

Programs

  • Mathematica
    nn=10;f[x_]:=Sum[Sum[Binomial[n,k]2^(k(n-k)),{k,0,n}]x^n/n!,{n,0,nn}];Range[0,nn]!CoefficientList[Series[Log[f[x]]+1,{x,0,nn}],x] (* Geoffrey Critzer, Sep 05 2013 *)
  • PARI
    seq(n)={Vec(serlaplace(1 + log(serconvol(sum(j=0, n, x^j*2^binomial(j, 2)) + O(x*x^n), (sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n))^2))))} \\ Andrew Howroyd, Dec 03 2018

Formula

a(n) = m_n(2) using the functions defined in A002032. - Sean A. Irvine, May 29 2013
E.g.f.: log(A(x))+1 where A(x) is the e.g.f. for A047863. - Geoffrey Critzer, Sep 05 2013
Logarithmic transform of A047863. - Andrew Howroyd, Dec 03 2018

Extensions

Corrected and extended by Sean A. Irvine, May 29 2013
Name clarified by Andrew Howroyd, Dec 03 2018

A002028 Number of connected graphs on n labeled nodes, each node being colored with one of 3 colors, such that no edge joins nodes of the same color.

Original entry on oeis.org

1, 3, 6, 42, 618, 15990, 668526, 43558242, 4373213298, 677307561630, 162826875512646, 61183069270120842, 36134310487980825258, 33673533885068169649830, 49646105434209446798290206, 116002075479856331220877149042, 430053223599741677879550609246498, 2531493110297317758855120762121050990
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • R. C. Read, personal communication.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column k=3 of A322279.
Cf. A002031, A002032.

Programs

  • Mathematica
    f[{k_, r_, m_}]:= Binomial[m+r+k, k] Binomial[m+r, r] 2^(k r +k m + r m);
  a = Sum[Total[Map[f, Compositions[n, 3]]] x^n/n!, {n, 0, 20}];
  Range[0, 20]! CoefficientList[Series[Log[a]+1, {x, 0, 20}], x] (* Geoffrey Critzer, Jun 02 2011 *)
  • PARI
    seq(n)={Vec(serlaplace(1 + log(serconvol(sum(j=0, n, x^j*2^binomial(j, 2)) + O(x*x^n), (sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n))^3))))} \\ Andrew Howroyd, Dec 03 2018

Formula

E.g.f.: log(A(x))+1 where A(x) is the e.g.f. for A191371. - Geoffrey Critzer, Jun 02 2011
a(n) = m_n(3) using the functions defined in A002032. - Sean A. Irvine, May 29 2013
Logarithmic transform of A191371. - Andrew Howroyd, Dec 03 2018

A002029 Number of connected graphs on n labeled nodes, each node being colored with one of 4 colors, such that no edge joins nodes of the same color.

Original entry on oeis.org

1, 4, 12, 132, 3156, 136980, 10015092, 1199364852, 234207001236, 75018740661780, 39745330657406772, 35073541377640231092, 51798833078501480220756, 128412490016744675540378580, 535348496386845235339961362932, 3757366291145650829115977555259252
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • R. C. Read, personal communication.

Links

Crossrefs

Column k=4 of A322279.
Cf. A002032.

Programs

  • Mathematica
    m = 16;
serconv = (CoefficientList[Sum[x^j*2^Binomial[j, 2], {j, 0, m}] + O[x]^m, x]*CoefficientList[(Sum[x^j/(j!*2^Binomial[j, 2]), {j, 0, m}] + O[x]^m)^4, x]) . x^Range[0, m-1];
CoefficientList[1 + Log[serconv] + O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, Sep 04 2019, after Andrew Howroyd *)
  • PARI
    seq(n)={Vec(serlaplace(1 + log(serconvol(sum(j=0, n, x^j*2^binomial(j, 2)) + O(x*x^n), (sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n))^4))))} \\ Andrew Howroyd, Dec 03 2018

Formula

E.g.f.: log(b(x)+1)+1 where b(x) = 4 * e.g.f. of A000686. - Sean A. Irvine, May 27 2013
a(n) = m_n(4) using the functions defined in A002032. - Sean A. Irvine, May 29 2013
Logarithmic transform of A223887. - Andrew Howroyd, Dec 03 2018

Extensions

More terms from Sean A. Irvine, May 27 2013
Name clarified and offset corrected by Andrew Howroyd, Dec 03 2018

A002030 Number of connected graphs on n labeled nodes, each node being colored with one of 5 colors, such that no edge joins nodes of the same color.

Original entry on oeis.org

1, 5, 20, 300, 9980, 616260, 65814020, 11878194300, 3621432947180, 1880516646144660, 1678121372919602420, 2590609089652498130700, 6947580541943715645962780, 32448510765823652400410879460, 264301377639329321236008592510820
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • R. C. Read, personal communication.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column k=5 of A322279.
Cf. A002032.

Programs

  • Mathematica
    m = 15;
serconv = (CoefficientList[Sum[x^j*2^Binomial[j, 2], {j, 0, m}] + O[x]^m, x]*CoefficientList[(Sum[x^j/(j!*2^Binomial[j, 2]), {j, 0, m}] + O[x]^m)^5, x]) . x^Range[0, m-1];
CoefficientList[1 + Log[serconv] + O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, Sep 04 2019, after Andrew Howroyd *)
  • PARI
    seq(n)={Vec(serlaplace(1 + log(serconvol(sum(j=0, n, x^j*2^binomial(j, 2)) + O(x*x^n), (sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n))^5))))} \\ Andrew Howroyd, Dec 03 2018

Formula

E.g.f.: log(B(x)+1) where B(x) = Sum_{n>=0} b(n)x^n/n! and b(n) = Sum_{j=0..n} C(n, j)*2^(j*(n-j)+2)*A000686(j). - Sean A. Irvine, May 27 2013
a(n) = m_n(5) using the functions defined in A002032. - Sean A. Irvine, May 29 2013

Extensions

More terms from Sean A. Irvine, May 27 2013
Name clarified and offset corrected by Andrew Howroyd, Dec 03 2018
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.