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 1-7 of 7 results.

A322280 Array read by antidiagonals: T(n,k) is the number of graphs on n labeled nodes, each node being colored with one of k colors, where no edge connects two nodes of the same color.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 15, 26, 1, 0, 1, 5, 28, 123, 162, 1, 0, 1, 6, 45, 340, 1635, 1442, 1, 0, 1, 7, 66, 725, 7108, 35043, 18306, 1, 0, 1, 8, 91, 1326, 20805, 254404, 1206915, 330626, 1, 0, 1, 9, 120, 2191, 48486, 1058885, 15531268, 66622083, 8488962, 1, 0
Offset: 0

Views

Author

Andrew Howroyd, Dec 01 2018

Keywords

Comments

Not all colors need to be used.

Examples

			Array begins:
===============================================================
n\k| 0 1      2        3          4           5           6
---+-----------------------------------------------------------
0  | 1 1      1        1          1           1           1 ...
1  | 0 1      2        3          4           5           6 ...
2  | 0 1      6       15         28          45          66 ...
3  | 0 1     26      123        340         725        1326 ...
4  | 0 1    162     1635       7108       20805       48486 ...
5  | 0 1   1442    35043     254404     1058885     3216486 ...
6  | 0 1  18306  1206915   15531268    95261445   386056326 ...
7  | 0 1 330626 66622083 1613235460 15110296325 83645197446 ...
...

Links

Crossrefs

Columns k=0..4 are A000007, A000012, A047863, A191371, A223887.
Main diagonal gives A372920.
Cf. A058843, A058875, A322278, A322279.

Programs

  • Mathematica
    nmax = 10;
T[n_, k_] := n!*2^Binomial[n, 2]*SeriesCoefficient[Sum[ x^i/(i!* 2^Binomial[i, 2]), {i, 0, nmax}]^k, {x, 0, n}];
Table[T[n - k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Sep 23 2019 *)
  • PARI
    M(n)={
  my(p=sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n));
  my(q=sum(j=0, n, x^j*j!*2^binomial(j, 2)) + O(x*x^n));
  matconcat([1, Mat(vector(n, k, Col(serconvol(q, p^k))))]);
}
my(T=M(7)); for(n=1, #T, print(T[n,]))

Formula

T(n,k) = n!*2^binomial(n,2) * [x^n](Sum_{i>=0} x^i/(i!*2^binomial(i,2)))^k.
T(n,k) = Sum_{j=0..k} binomial(k,j)*j!*A058843(n,j).

A322278 Triangle read by rows: T(n,k) is the number of k-colored connected graphs on n labeled nodes up to permutation of the colors.

Original entry on oeis.org

1, 0, 1, 0, 3, 4, 0, 19, 84, 38, 0, 195, 2470, 3140, 728, 0, 3031, 108390, 307390, 186360, 26704, 0, 67263, 7192444, 42747460, 52630060, 18926544, 1866256, 0, 2086099, 726782784, 9030799218, 20784069600, 14401134944, 3463311488, 251548592
Offset: 1

Views

Author

Andrew Howroyd, Dec 01 2018

Keywords

Comments

Equivalently, the number of ways to choose a stable partition of a simple connected graph on n labeled nodes with k parts. See A322064 for the definition of stable partition.

Examples

			Triangle begins:
  1;
  0,     1;
  0,     3,       4;
  0,    19,      84,       38;
  0,   195,    2470,     3140,      728;
  0,  3031,  108390,   307390,   186360,    26704;
  0, 67263, 7192444, 42747460, 52630060, 18926544, 1866256;
  ...

Links

Crossrefs

Row sums are A322064.
Columns k=2..4 are A001832(for n > 1), A322330, A322331.
Right diagonal is A001187.
Cf. A058843, A058875, A322279, A322280.

Programs

  • PARI
    M(n, K=n)={
  my(p=sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n));
  my(q=sum(j=0, n, x^j*2^binomial(j, 2)) + O(x*x^n));
  my(W=vector(K, k, Col(serlaplace(log(serconvol(q, p^k))))));
  Mat(vector(K, k, sum(i=1, k, (-1)^(k-i)*binomial(k,i)*W[i])/k!));
}
my(T=M(7)); for(n=1, #T, print(T[n, 1..n]))

Formula

T(n,k) = (1/k!)*Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*A322279(n,j).

A002032 Number of n-colored connected graphs on n labeled nodes.

Original entry on oeis.org

1, 1, 2, 24, 912, 87360, 19226880, 9405930240, 10142439229440, 24057598104207360, 125180857812868300800, 1422700916050060841779200, 35136968950395142864227532800, 1876028272361273394915958613606400, 215474119792145796020405035320528076800
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Every connected graph on n nodes can be colored with n colors in exactly n! ways, so this sequence is just n! * A001187(n). - Andrew Howroyd, Dec 03 2018

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. A002027. A002028, A002029, A002030, A002031.
Cf. A001187, A322278, A322279, A322280.

Programs

  • Mathematica
    (* b = A001187 *) b[n_] := b[n] = If[n == 0, 1, 2^(n(n-1)/2) - Sum[k* Binomial[n, k]*2^((n-k)(n-k-1)/2)*b[k], {k, 1, n-1}]/n];
a[n_] := n! b[n];
Array[a, 14] (* Jean-François Alcover, Aug 16 2019, using Alois P. Heinz's code for A001187 *)
  • PARI
    seq(n) = {Vec(serlaplace(serlaplace(1 + log(sum(k=0, n, 2^binomial(k, 2)*x^k/k!, O(x*x^n))))))} \\ Andrew Howroyd, Dec 03 2018

Formula

a(n) = n!*A001187(n). - Andrew Howroyd, Dec 03 2018
Define M_0(k)=1, M_n(0)=0, M_n(k) = Sum_{r=0..n} C(n,r)*2^(r*(n-r))*M_r(k-1) [M_n(k) = A322280(n,k)], m_n(k) = M_n(k) -Sum_{r=1..n-1} C(n-1,r-1)*m_r(k)*M_{n-r}(k) [m_n(k) = A322279(n,k)], f_n(k) = Sum_{r=1..k} (-1)^(k-r)*C(k,r)*m_n(r). This sequence gives a(n) = f_n(n). - Sean A. Irvine, May 29 2013, edited Andrew Howroyd, Dec 03 2018
The above formula is referenced by sequences A002027-A002030, A002031. - Andrew Howroyd, Dec 03 2018

Extensions

More terms from Sean A. Irvine, May 29 2013
Name clarified by Andrew Howroyd, Dec 03 2018
a(0)=1 prepended by Andrew Howroyd, Jan 05 2024

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 1-7 of 7 results.

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

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