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.

A182368 Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the square grid graph G_(n,n), highest powers first.

Original entry on oeis.org

1, 0, 1, -4, 6, -3, 0, 1, -12, 66, -216, 459, -648, 594, -323, 79, 0, 1, -24, 276, -2015, 10437, -40614, 122662, -292883, 557782, -848056, 1022204, -960627, 682349, -346274, 112275, -17493, 0, 1, -40, 780, -9864, 90798, -647352, 3714180, -17590911, 69997383
Offset: 1

Views

Author

Alois P. Heinz, Apr 26 2012

Keywords

Comments

The square grid graph G_(n,n) has n^2 = A000290(n) vertices and 2*n*(n-1) = A046092(n-1) edges. The chromatic polynomial of G_(n,n) has n^2+1 = A002522(n) coefficients.

Examples

			3 example graphs:                          o---o---o
.                                          |   |   |
.                             o---o        o---o---o
.                             |   |        |   |   |
.                o            o---o        o---o---o
Graph:        G_(1,1)        G_(2,2)        G_(3,3)
Vertices:        1              4              9
Edges:           0              4             12
The square grid graph G_(2,2) is the cycle graph C_4 with chromatic polynomial q^4 -4*q^3 +6*q^2 -3*q => row 2 = [1, -4, 6, -3, 0].
Triangle T(n,k) begins:
  1,    0;
  1,   -4,     6,      -3,        0;
  1,  -12,    66,    -216,      459,       -648,         594, ...
  1,  -24,   276,   -2015,    10437,     -40614,      122662, ...
  1,  -40,   780,   -9864,    90798,    -647352,     3714180, ...
  1,  -60,  1770,  -34195,   486210,   -5421612,    49332660, ...
  1,  -84,  3486,  -95248,  1926585,  -30755376,   403410654, ...
  1, -112,  6216, -227871,  6205479, -133865298,  2382122274, ...
  1, -144, 10296, -487280, 17169852, -480376848, 11114098408, ...
  ...

Links

Crossrefs

Columns 0, 1 give: A000012, (-1)*A046092(n-1).
Sums of absolute values of row elements give: A080690(n).
Cf. A000290, A002522, A182406, A185442, A193233, A193277, A193283.

Programs

  • Mathematica
    Reverse /@ CoefficientList[Table[ChromaticPolynomial[GridGraph[{n, n}], x], {n, 5}], x] // Flatten (* Eric W. Weisstein, May 01 2017 *)

A182797 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the k X k X k triangular grid.

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 0, 0, 6, 4, 0, 0, 6, 24, 5, 0, 0, 6, 192, 60, 6, 0, 0, 6, 2112, 1620, 120, 7, 0, 0, 6, 32640, 98820, 7680, 210, 8, 0, 0, 6, 718080, 13638780, 1574400, 26250, 336, 9, 0, 0, 6, 22665216, 4260983940, 1034019840, 13676250, 72576, 504, 10
Offset: 1

Views

Author

Alois P. Heinz, Dec 02 2010

Keywords

Comments

The k X k X k triangular grid has k rows with i vertices in row i. Each vertex is connected to the neighbors in the same row and up to two vertices in each of the neighboring rows. The graph has A000217(k) vertices and 3*A000217(k-1) edges altogether.
The coefficients of the chromatic polynomials for the column sequences are given by the rows of A193283. - Georg Fischer, Jul 31 2023

Examples

			Square array A(n,k) begins:
  1,   0,    0,       0,          0,             0,  ...
  2,   0,    0,       0,          0,             0,  ...
  3,   6,    6,       6,          6,             6,  ...
  4,  24,  192,    2112,      32640,        718080,  ...
  5,  60, 1620,   98820,   13638780,    4260983940,  ...
  6, 120, 7680, 1574400, 1034019840, 2175789895680,  ...

Links

Crossrefs

Columns k=1-11 give: A000027, A007531, A182788, A182789, A182790, A182791, A182792, A182793, A182794, A182795, A182796.
Rows n=1-10 give: A000007(k-1), A000038(k-1), A040006(k-1), A182798, A153467*4, A153468*5, A153469*6, A153470*7, A153471*8, A153472*9, A153473*10.
Cf. A000217, A193283.

A212084 Triangle T(n,k), n>=0, 0<=k<=2n, read by rows: row n gives the coefficients of the chromatic polynomial of the complete bipartite graph K_(n,n), highest powers first.

Original entry on oeis.org

1, 1, -1, 0, 1, -4, 6, -3, 0, 1, -9, 36, -75, 78, -31, 0, 1, -16, 120, -524, 1400, -2236, 1930, -675, 0, 1, -25, 300, -2200, 10650, -34730, 75170, -102545, 78610, -25231, 0, 1, -36, 630, -6915, 52080, -279142, 1074822, -2942445, 5552680, -6796926, 4787174
Offset: 0

Views

Author

Alois P. Heinz, Apr 30 2012

Keywords

Comments

The complete bipartite graph K_(n,n) has 2n vertices and n^2 = A000290(n) edges. The chromatic polynomial of K_(n,n) has 2n+1 = A005408(n) coefficients.

Examples

			3 example graphs:                     +-----------+
.                 o        o   o      o   o   o   |
.                 |        |\ /|      |\ /|\ /|\ /
.                 |        | X |      | X | X | X
.                 |        |/ \|      |/ \|/ \|/ \
.                 o        o   o      o   o   o   |
.                                     +-----------+
Graph:         K_(1,1)    K_(2,2)      K_(3,3)
Vertices:         2          4            6
Edges:            1          4            9
The complete bipartite graph K_(2,2) is the cycle graph C_4 with chromatic polynomial q^4 -4*q^3 +6*q^2 -3*q => row 2 = [1, -4, 6, -3, 0].
Triangle T(n,k) begins:
  1;
  1,  -1,   0;
  1,  -4,   6,    -3,     0;
  1,  -9,  36,   -75,    78,     -31,       0;
  1, -16, 120,  -524,  1400,   -2236,    1930,     -675, ...
  1, -25, 300, -2200, 10650,  -34730,   75170,  -102545, ...
  1, -36, 630, -6915, 52080, -279142, 1074822, -2942445, ...
  ...

Links

Crossrefs

Columns k=0-2 give: A000012, (-1)*A000290, A083374.
Row sums and last elements of rows give: A000007.
Row lengths give: A005408.
Sums of absolute values of row elements give: A048163(n+1).
T(n,2n-1) = (-1)*A092552(n).
Cf. A008277, A212085, A182368, A185442, A193233, A193277, A193283, A266695, A266972.

Programs

  • Maple
    P:= n-> add(Stirling2(n, k) *mul(q-i, i=0..k-1) *(q-k)^n, k=0..n):
T:= n-> seq(coeff(P(n), q, 2*n-k), k=0..2*n):
seq(T(n), n=1..8);

Formula

T(n,k) = [q^(2n-k)] Sum_{j=0..n} (q-j)^n * S2(n,j) * Product_{i=0..j-1} (q-i).

Extensions

T(0,0)=1 prepended by Alois P. Heinz, May 03 2024

A182283 Number of triangular n X n X n arrays colored with integers 0 upwards introduced in row major order, with no element equal to any neighbor.

Original entry on oeis.org

1, 1, 15, 3429, 18172005, 3030361658604, 20538495213667066533, 7069329642959332230532689983, 150574890630606350105309341350824904669, 237075065354315062816111131522815337395866137560373, 32430625006159571889921247597353572731767630164652210957593666925
Offset: 1

Views

Author

R. H. Hardin, Apr 23 2012

Keywords

Examples

			Some solutions for n=4
.....0........0........0........0........0........0........0........0
....1.2......1.2......1.2......1.2......1.2......1.2......1.2......1.2
...2.0.3....3.4.0....2.0.1....0.3.4....3.4.0....3.4.0....3.4.5....3.4.0
..3.1.4.1..4.2.5.4..3.1.3.0..4.5.2.5..0.5.3.6..4.5.3.6..5.0.2.1..1.2.3.5

Crossrefs

Cf. A182797, A193283.

Extensions

a(7)-a(11) from Andrew Howroyd, Apr 23 2018

A295190 Chromatic invariant of the n-triangular grid graph.

Original entry on oeis.org

1, 1, 1, 5, 97, 6739, 1611097, 1295101469, 3449859538455, 30155591559236245, 859063676925680110319, 79361177641450830904290293
Offset: 0

Views

Author

Eric W. Weisstein, Nov 16 2017

Keywords

Links

Crossrefs

Cf. A182283, A182797, A193283.

Extensions

a(8)-a(11) from Andrew Howroyd, Apr 23 2018
Showing 1-5 of 5 results.

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