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.

User: Andrew Howroyd

Andrew Howroyd's wiki page.

Andrew Howroyd has authored 1185 sequences. Here are the ten most recent ones:

A384968 Triangle read by rows: T(n,k) is the number of proper vertex colorings of the n-complete bipartite graph using exactly k interchangeable colors, 2 <= k <= 2*n.

Original entry on oeis.org

1, 1, 2, 1, 1, 6, 11, 6, 1, 1, 14, 61, 86, 50, 12, 1, 1, 30, 275, 770, 927, 530, 150, 20, 1, 1, 62, 1141, 5710, 12160, 12632, 6987, 2130, 355, 30, 1, 1, 126, 4571, 38626, 134981, 228382, 209428, 110768, 34902, 6580, 721, 42, 1, 1, 254, 18061, 248766, 1367310, 3553564, 4989621, 4093126, 2061782, 655788, 132958, 16996, 1316, 56, 1
Offset: 1

Views

Author

Andrew Howroyd, Jun 18 2025

Keywords

Comments

Permuting the colors does not change the coloring. T(n,k) is the number of ways to partition the vertices into k independent sets.

Examples

			Triangle begins (n >= 1, k >= 2):
  1;
  1,  2,    1;
  1,  6,   11,    6,     1;
  1, 14,   61,   86,    50,    12,    1;
  1, 30,  275,  770,   927,   530,  150,   20,   1;
  1, 62, 1141, 5710, 12160, 12632, 6987, 2130, 355, 30, 1;
  ...

Links

Crossrefs

Row sums are A001247.
Columns k=2..5 are A000012, A000918, A384980, A384981.
Cf. A008277, A212084, A274310.

Programs

  • PARI
    T(n,k) = sum(j=1, k-1, stirling(n,j,2)*stirling(n,k-j,2))
for(n=1, 7, print(vector(2*n-1,k,T(n,k+1))))

Formula

T(n,k) = Sum_{j=1..k-1} Stirling2(n,j)*Stirling2(n,k-j).
T(n,k) = A274310(2*n-1, k-1).

A384850 Triangle read by rows: T(n,k) is the number of unsensed simple planar maps with n edges and k vertices, 1 <= k <= n+1.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 2, 3, 0, 0, 0, 1, 7, 6, 0, 0, 0, 1, 7, 22, 12, 0, 0, 0, 0, 5, 42, 76, 27, 0, 0, 0, 0, 2, 49, 237, 271, 65, 0, 0, 0, 0, 1, 35, 442, 1293, 1001, 175, 0, 0, 0, 0, 0, 18, 510, 3539, 6757, 3765, 490
Offset: 0

Views

Author

Andrew Howroyd, Jun 13 2025

Keywords

Comments

The planar maps considered here are connected.
The initial terms of this sequence can be computed using the tool "plantri", in particular the command "./plantri -u -v -c1 -p [n]" will compute values for a column.

Examples

			Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 2;
  0, 0, 0, 2, 3;
  0, 0, 0, 1, 7,  6;
  0, 0, 0, 1, 7, 22,  12;
  0, 0, 0, 0, 5, 42,  76,   27;
  0, 0, 0, 0, 2, 49, 237,  271,   65;
  0, 0, 0, 0, 1, 35, 442, 1293, 1001, 175;
  ...

Links

Crossrefs

Row sums are A006395.
Column sums are A372892.
Main diagonal is A006082.
Subdiagonal is A384967.
Cf. A054923 (graphs), A277741 (not necessarily simple), A342060 (2-connected), A212438 (3-connected), A384963 (version by number of vertices then faces).

A384963 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of connected simple planar graphs with n nodes and k faces, n >= 1, k=1..max(1,2*n-4).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 3, 7, 7, 5, 2, 1, 6, 22, 42, 49, 35, 18, 5, 2, 12, 76, 237, 442, 510, 412, 218, 84, 18, 5, 27, 271, 1293, 3539, 6205, 7482, 6318, 3833, 1623, 485, 88, 14, 65, 1001, 6757, 25842, 63254, 106985, 129782, 115988, 76582, 37421, 13111, 3228, 489, 50
Offset: 1

Views

Author

Andrew Howroyd, Jun 13 2025

Keywords

Comments

Equivalently, T(n,k) is the number of unsensed simple planar maps with n vertices and k faces.
The number of edges is n+k-2.
Terms of this sequence can be computed using the tool "plantri". The expanded reference gives rows 1..14 of this table.

Examples

			Triangle begins:
   1;
   1;
   1,   1;
   2,   2,    1,    1,
   3,   7,    7,    5,    2,    1;
   6,  22,   42,   49,   35,   18,    5,    2;
  12,  76,  237,  442,  510,  412,  218,   84,   18,   5;
  27, 271, 1293, 3539, 6205, 7482, 6318, 3833, 1623, 485, 88, 14;
  ...

Links

Crossrefs

Row sums are A372892.
Antidiagonal sums are A006395.
Columns 1..2 are A006082, A384967.
Cf. A277741 (not necessarily simple), A342060 (2-connected), A212438 (3-connected), A384850 (version by number of edges then vertices), A384964 (sensed version).

A384964 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of connected simple planar graphs with n nodes and k faces up to orientation preserving isomorphisms, n >= 1, k=1..max(1,2*n-4).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 3, 8, 8, 6, 2, 1, 6, 29, 60, 73, 52, 25, 6, 2, 14, 113, 388, 768, 903, 728, 379, 136, 26, 6, 34, 444, 2303, 6584, 11782, 14321, 12113, 7298, 3048, 872, 147, 17, 95, 1763, 12650, 49806, 123547, 210314, 255884, 228807, 150929, 73428, 25536, 6142, 892, 73
Offset: 1

Views

Author

Andrew Howroyd, Jun 13 2025

Keywords

Comments

Equivalently, T(n,k) is the number of sensed simple planar maps with n vertices and k faces.
The number of edges is n+k-2.
Terms of this sequence can be computed using the tool "plantri". The expanded reference gives rows 1..14 of this table.

Examples

			Triangle begins:
   1;
   1;
   1,   1,
   2,   2,    1,    1,
   3,   8,    8,    6,     2,     1,
   6,  29,   60,   73,    52,    25,     6,    2,
  14, 113,  388,  768,   903,   728,   379,  136,   26,   6,
  34, 444, 2303, 6584, 11782, 14321, 12113, 7298, 3048, 872, 147, 17;
  ...

Links

Crossrefs

Row sums are A384965.
Antidiagonal sums are A006394.
Columns 1..2 are A002995, A384966.
Cf. A379430 (not necessarily simple), A342059 (2-connected), A239893 (3-connected), A384963 (unsensed).

A384965 Number of sensed simple planar maps with n vertices.

Original entry on oeis.org

1, 1, 2, 6, 28, 253, 3461, 58963, 1139866, 23952568, 534729502, 12511055327, 303919972592, 7613826460120
Offset: 1

Views

Author

Andrew Howroyd, Jun 13 2025

Keywords

Comments

A simple planar map is a planar map without loops or parallel edges.

Links

Crossrefs

Row sums of A384964.
Cf. A006394 (with n edges), A372892 (unsensed version).

A384966 Number of sensed simple planar maps with n vertices and 2 faces.

Original entry on oeis.org

0, 0, 1, 2, 8, 29, 113, 444, 1763, 6951, 27395, 107672, 422330, 1654180, 6472518, 25308760, 98923442, 386589398, 1510737079, 5904291401, 23079308104, 90236258057, 352908128341, 1380632536468, 5403055984114, 21152009997924, 82835786189975, 324518950873991, 1271797441923614, 4985982054721119
Offset: 1

Views

Author

Andrew Howroyd, Jun 14 2025

Keywords

Comments

In other words, a(n) is the number of embeddings on the sphere of connected simple unicyclic planar graphs with n nodes up to orientation preserving isomorphisms.

Links

Crossrefs

Column 2 of A384964.
Cf. A001429, A006078 (cycle is loop), A007595 (cycle is digon), A380237 (not necessarily simple), A384967 (unsensed version)..

Programs

  • PARI
    seq(n)={my(c(d)=(1-sqrt(1-4*x^d + O(x*x^(n+d))))/(2*x^d)); Vec(1/(1 - x*c(2)) - x*(c(1)^2 + c(2)) - x^2*(c(1)^4 + 3*c(2)^2)/2 - 1 - sum(k=1, n, log(2 - c(k))*eulerphi(k)/k), -n)/2}

Formula

a(n) = A380237(n) - A007595(n) - A006078(n).

A384967 Number of unsensed simple planar maps with n vertices and 2 faces.

Original entry on oeis.org

0, 0, 1, 2, 7, 22, 76, 271, 1001, 3765, 14381, 55450, 214880, 835663, 3255652, 12698352, 49559793, 193513944, 755852101, 2953214386, 11541989533, 45123241746, 176465152051, 690340349398, 2701579878022, 10576116931462, 41418132927403, 162259989848094, 635899817853002, 2492993368347594
Offset: 1

Views

Author

Andrew Howroyd, Jun 15 2025

Keywords

Comments

In other words, a(n) is the number of embeddings on the sphere of connected simple unicyclic planar graphs with n nodes.

Crossrefs

Column 2 of A384963.
Also subdiagonal of A379430.
Cf. A001429, A006081 (cycle is loop), A380239 (not necessarily simple), A384966 (sensed version).

Programs

  • PARI
    G1(n)={my(g=(1-sqrt(1-4*x^2 + O(x^(n+2))))/(2*x^2)); ((1 + x/(1-x-x^2*g)^2)^2/(1 - x^2*g^2) - 1)/2 + 1/(1 - x*g) - 1 - x*(g^2/(1 - x*g)^2 + g) - x^2*(g^4/(1 - x*g)^4 + 3*g^2)/2}
G2(n)={my(c(d)=(1-sqrt(1-4*x^d + O(x*x^(n+d))))/(2*x^d)); sum(k=1, n, my(m=1+k%2); -(log(2 - c(k)) + log(1 - x^k*c(m*k)^(2/m)))*eulerphi(k)/k, O(x*x^n)) - x*(c(1)^2 + c(2)) - x^2*(c(1)^4 + 3*c(2)^2)/2}
seq(n)={Vec(G1(n)+G2(n), -n)/4}

A384843 Wiener index of the n-Dorogovtsev-Goltsev-Mendes graph.

Original entry on oeis.org

1, 3, 21, 204, 2130, 22245, 229119, 2325966, 23319708, 231384327, 2276119977, 22228324368, 215745006246, 2082918495849, 20017195390995, 191593142789010, 1827283815276144, 17372064324294411, 164687169445632573, 1557231841690641492, 14690512431146615802
Offset: 0

Views

Author

Andrew Howroyd, Jun 10 2025

Keywords

Links

Crossrefs

Cf. A384844.

Programs

  • Mathematica
    A384843[n_] := If[n == 0, 1, 3*(1 + 10*3^(n - 2) + 3*(4*n + 7)*9^(n - 2))/8];
Array[A384843, 25, 0] (* Paolo Xausa, Jun 18 2025 *)
  • PARI
    a(n) = if(n == 0, 1, 3*(1 + 10*3^(n-2) + 3*(4*n + 7)*9^(n-2))/8)

Formula

a(n) = 3*(1 + 10*3^(n-2) + 3*(4*n + 7)*9^(n-2))/8 for n > 0.
a(n) = 22*a(n-1) - 156*a(n-2) + 378*a(n-3) - 243*a(n-4) for n > 4.
a(n) = Sum_{k=1..n} k*A384844(n,k) for n > 0.
G.f.: 1 + 3*x*(1 - 15*x + 70*x^2 - 72*x^3)/((1 - x)*(1 - 3*x)*(1 - 9*x)^2).
E.g.f.: (8 + 27*exp(x) + 30*exp(3*x) + exp(9*x)*(7 + 36*x))/72. - Stefano Spezia, Jun 14 2025

A384844 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the n-Dorogovtsev-Goltsev-Mendes graph.

Original entry on oeis.org

3, 9, 6, 27, 57, 21, 81, 351, 369, 60, 243, 1806, 3582, 1716, 156, 729, 8472, 26346, 24216, 6648, 384, 2187, 37683, 165375, 241032, 128880, 22896, 912, 6561, 162177, 938907, 1946676, 1670280, 584784, 72624, 2112, 19683, 683112, 4979928, 13697148, 16889340, 9580368, 2366256, 216768, 4800
Offset: 1

Views

Author

Andrew Howroyd, Jun 10 2025

Keywords

Examples

			Triangle begins:
     3;
     9,     6;
    27,    57,     21;
    81,   351,    369,     60;
   243,  1806,   3582,   1716,    156;
   729,  8472,  26346,  24216,   6648,   384;
  2187, 37683, 165375, 241032, 128880, 22896, 912;
  ...

Links

Crossrefs

Main diagonal is A113070(n-1) for n > 1.
Column 1 is A000244.
Cf. A384843.

Programs

  • PARI
    T(n)={ my(c=x^2*y/((1 - x)*(1 - 3*x + 2*(1 - y)*x^2)) + O(x*x^n), b=(1-2*x)*c/x, g = y*(1+b+2*c) + serconvol(b + c, b + c + y*c) + serconvol(y*c, b + 2*c)); [Vecrev(p/y)|p<-Vec(3*g/(1 - 3*x))]}
{ foreach(T(10), row, print(row)) }
  • PARI
    T(n)={my(g=3*(1 - 2*(3 + y)*x + 3*(3 - y + y^2)*x^2 - 4*(1 - y)^2*x^3)/((1 - x)*(1 - 3*x)*(1 - (5 + 4*y)*x + 4*(1 - y)^2*x^2))); [Vecrev(p)|p<-Vec(g + O(x^n))]}
{ foreach(T(10), row, print(row)) }

Formula

G.f.: 3*x*y*(1 - 2*(3 + y)*x + 3*(3 - y + y^2)*x^2 - 4*(1 - y)^2*x^3)/((1 - x)*(1 - 3*x)*(1 - (5 + 4*y)*x + 4*(1 - y)^2*x^2)).
A384843(n) = Sum_{k=1..n} k*T(n,k).

A384849 Triangle read by rows: T(n,k) is the number of simple graphs on n unlabeled nodes with degeneracy k, 0 <= k < n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 4, 1, 1, 9, 18, 5, 1, 1, 19, 85, 43, 7, 1, 1, 36, 471, 442, 85, 8, 1, 1, 75, 3378, 6979, 1758, 144, 10, 1, 1, 152, 31782, 166258, 70811, 5421, 231, 11, 1, 1, 328, 385205, 5892753, 5164116, 547170, 15239, 342, 13, 1
Offset: 1

Views

Author

Andrew Howroyd, Jun 10 2025

Keywords

Examples

			Triangle begins:
  1;
  1,   1;
  1,   2,      1;
  1,   5,      4,       1;
  1,   9,     18,       5,       1;
  1,  19,     85,      43,       7,      1;
  1,  36,    471,     442,      85,      8,     1;
  1,  75,   3378,    6979,    1758,    144,    10,   1;
  1, 152,  31782,  166258,   70811,   5421,   231,  11,  1;
  1, 328, 385205, 5892753, 5164116, 547170, 15239, 342, 13, 1;
  ...

Links

Crossrefs

Row sums are A000088.
Cf. A005195, A352067 (connected case).

Formula

T(n,0) = T(n,n-1) = 1.
T(n,1) = A005195(n) - 1.

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

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