A334279 -id:A334279 - cp's OEIS frontend

A334278 Irregular table read by rows: T(n, k) is the coefficient of x^k in the chromatic polynomial of the cubical graph Q_n, 0 <= k <= 2^n.

0, 1, 0, -1, 1, 0, -3, 6, -4, 1, 0, -133, 423, -572, 441, -214, 66, -12, 1, 0, -3040575, 14412776, -31680240, 43389646, -41821924, 30276984, -17100952, 7701952, -2794896, 818036, -191600, 35264, -4936, 496, -32, 1

Offset: 0

Peter Kagey, Apr 21 2020

The sums of the absolute values of the entries in each row gives A334247, the number of acyclic orientations of edges of the n-cube.

			Table begins:
n/k| 0     1    2     3    4     5   6    7  8
---+-------------------------------------------
  0| 0,    1
  1| 0,   -1,   1
  2| 0,   -3,   6,   -4,   1
  3| 0, -133, 423, -572, 441, -214, 66, -12, 1

Peter Kagey, Table of n, a(n) for n = 0..68 (rows 0..5; row 5 from Andrew Howroyd's file)
Andrew Howroyd, Chromatic Polynomials of Hypercubes Q_0 to Q_5
Eric Weisstein's World of Mathematics, Chromatic Polynomial.
Eric Weisstein's World of Mathematics, Hypercube Graph.

Cf. A296914 is the reverse of row 3.
Cf. A334279 is analogous for the n-dimensional cross-polytope, the dual of the n-cube.
Cf. A001787, A002378, A091940, A140986, A158348.
Cf. A334159, A334247, A334358.

with(GraphTheory): with(SpecialGraphs):
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
    ChromaticPolynomial(HypercubeGraph(n), x)):
seq(T(n), n=0..4);  # Alois P. Heinz, Jan 14 2025

T[n_, k_] := Coefficient[ChromaticPolynomial[HypercubeGraph[n], x], x, k]

T(n,0) = 0.

T(n,k) = Sum_{i=1..2^n}, Stirling1(i,k) * A334159(n,i). - Andrew Howroyd, Apr 25 2020

A334281 Number of n-colorings of the vertices of the 4-dimensional cross polytope such that no two adjacent vertices have the same color.

Original entry on oeis.org

0, 0, 0, 0, 24, 600, 7560, 61320, 351120, 1515024, 5266800, 15531120, 40308840, 94534440, 204228024, 412284600, 786283680, 1428742560, 2490276960, 4186173024, 6816915000, 10793253240, 16666437480, 25164280680, 37233759024, 54090894000, 77278702800

Offset: 0

Peter Kagey, Apr 21 2020

The 4-dimensional cross-polytope is sometimes called the 16-cell. It is one of the six convex regular 4-polytopes.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Chromatic Polynomial
Wikipedia, Cross-polytope
Wikipedia, Turán graph
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A091940 (2-dimensional), A115400 (3-dimensional).

Cf. A334279.

concat([0,0,0,0], Vec(24*x^4*(1 + 16*x + 126*x^2 + 536*x^3 + 1001*x^4) / (1 - x)^9 + O(x^30))) \\ Colin Barker, Apr 22 2020

a(n) = n*(n - 1)*(n - 2)*(n - 3)*(465 - 392n + 125n^2 - 18n^3 + n^4).
a(n) = -2790n + 7467n^2 - 7852n^3 + 4300n^4 - 1346n^5 + 244n^6 - 24n^7 + n^8.
From Colin Barker, Apr 22 2020: (Start)
G.f.: 24*x^4*(1 + 16*x + 126*x^2 + 536*x^3 + 1001*x^4) / (1 - x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>8.
(End)

A342088 Triangle read by rows: T(n,k) is the number of n-colorings of the vertices of the k-dimensional cross polytope such that no two adjacent vertices have the same color. 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 9, 18, 6, 1, 16, 84, 96, 24, 1, 25, 260, 780, 600, 120, 1, 36, 630, 4080, 7560, 4320, 720, 1, 49, 1302, 15330, 61320, 78120, 35280, 5040, 1, 64, 2408, 45696, 351120, 913920, 866880, 322560, 40320

Offset: 0

Peter Kagey, Feb 27 2021

			Triangle begins:
  n\k| 0   1     2      3       4       5       6       7      8
  ---+----------------------------------------------------------
   0 | 1
   1 | 1,  1
   2 | 1,  4,    2
   3 | 1,  9,   18,     6
   4 | 1, 16,   84,    96,     24
   5 | 1, 25,  260,   780,    600,    120
   6 | 1, 36,  630,  4080,   7560,   4320,    720
   7 | 1, 49, 1302, 15330,  61320,  78120,  35280,   5040
   8 | 1, 64, 2408, 45696, 351120, 913920, 866880, 322560, 40320

Peter Kagey, Rows n = 0..100, flattened
Eric Weisstein's World of Mathematics, Chromatic Polynomial
Eric Weisstein's World of Mathematics, Cocktail Party Graph
Wikipedia, Cross-polytope
Wikipedia, Turán graph

Cf. A000012 (k=0), A000290 (k=1), A091940 (k=2), A115400 (k=3), A334281 (k=4), A342073 (k=5), A342074 (k=6), A342075 (k=7).

Cf. A334279.

T[n_, k_] := Sum[n! k!/((n - k - j)! (k - j)! j!), {j, 0, k}]

T(n,n) = n!.
T(n,k) = Sum_{i=0..2*k} A334279(k,i)*n^i.
T(n,k) = n*T(n-1,k-1) + n*(n-1)*T(n-2,k-1).
T(n,k) = Sum_{j=0..k} n!k!/((n-k-j)!(k-j)!j!).

A342073 Number of n-colorings of the vertices of the 5-dimensional cross polytope such that no two adjacent vertices have the same color.

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 4320, 78120, 913920, 7575120, 46751040, 224587440, 881591040, 2946869640, 8659691040, 22915652760, 55611279360, 125508233760, 266320172160, 535945217760, 1030028705280, 1901347885080, 3386866301280, 5844714201480, 9803816225280

Offset: 0

Peter Kagey, Feb 27 2021

nonn

Peter Kagey, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Analogous for k-dimensional cross polytope: A091940 (k=2), A115400 (k=3), A334281 (k=4), A342074 (k=6), A342075 (k=7)

Cf. A334279, A342088.

p = ChromaticPolynomial[CompleteGraph[Table[2, 5]], x];
Table[p /. x -> n, {n, 0, 50}]

a(n) = -205056*n + 593016*n^2 - 698250*n^3 + 448015*n^4 - 175004*n^5 + 43608*n^6 - 6990*n^7 + 700*n^8 - 40*n^9 + n^10.
a(n) = (n - 4)*(n - 3)*(n - 2)*(n - 1)*n*(-8544 + 6909*n - 2240*n^2 + 365*n^3 - 30*n^4 + n^5).
a(n) = Sum_{i=1..10} A334279(5,i)*n^i.
From Chai Wah Wu, Jan 19 2024: (Start)
a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11) for n > 10.
G.f.: x^5*(-2170680*x^5 - 1145400*x^4 - 272400*x^3 - 37200*x^2 - 3000*x - 120)/(x - 1)^11. (End)

A342074 Number of n-colorings of the vertices of the 6-dimensional cross polytope such that no two adjacent vertices have the same color.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 720, 35280, 866880, 13849920, 158004000, 1347524640, 8866186560, 46496324160, 201705744240, 748737990000, 2444976293760, 7178449299840, 19276199691840, 47983899216960, 111920569776000, 246727594270080, 517702915311120, 1039979954779920

Offset: 0

Peter Kagey, Feb 27 2021

nonn

Peter Kagey, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Analogous for k-dimensional cross polytope: A091940 (k=2), A115400 (k=3), A334281 (k=4), A342073 (k=5), A342075 (k=7).

Cf. A334279, A342088.

p = ChromaticPolynomial[CompleteGraph[Table[2, 6]], x];
Table[p /. x -> n, {n, 0, 50}]

a(n) = -22852200*n + 70164670*n^2 - 89812001*n^3 + 64407806*n^4 - 29113410*n^5 + 8790285*n^6 - 1822164*n^7 + 260868*n^8 - 25405*n^9 + 1610*n^10 - 60*n^11 + n^12.
a(n) = (n - 5)*(n - 4)*(n - 3)*(n - 2)*(n - 1)*n*(190435 - 149879*n + 49144*n^2 - 8605*n^3 + 850*n^4 - 45*n^5 + n^6).
a(n) = Sum_{i=1..12} A334279(6,i)*n^i.
From Chai Wah Wu, Jan 19 2024: (Start)
a(n) = 13*a(n-1) - 78*a(n-2) + 286*a(n-3) - 715*a(n-4) + 1287*a(n-5) - 1716*a(n-6) + 1716*a(n-7) - 1287*a(n-8) + 715*a(n-9) - 286*a(n-10) + 78*a(n-11) - 13*a(n-12) + a(n-13) for n > 12.
G.f.: x^6*(-287250480*x^6 - 150137280*x^5 - 35996400*x^4 - 5126400*x^3 - 464400*x^2 - 25920*x - 720)/(x - 1)^13. (End)

A342075 Number of n-colorings of the vertices of the 7-dimensional cross polytope such that no two adjacent vertices have the same color.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 5040, 322560, 10342080, 216518400, 3261535200, 37026823680, 325474269120, 2264594492160, 12789814237200, 60389186457600, 245221330273920, 877374833287680, 2821277454690240, 8284633867238400, 22503569636419200, 57135310310453760

Offset: 0

Peter Kagey, Feb 27 2021

nonn

Peter Kagey, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Analogous for k-dimensional cross polytope: A091940 (k=2), A115400 (k=3), A334281 (k=4), A342073 (k=5), A342074 (k=6).

Cf. A334279, A342088.

p = ChromaticPolynomial[CompleteGraph[Table[2, 7]], x];
Table[p /. x -> n, {n, 0, 50}]

a(n) = -3597143040*n + 11590795728*n^2 - 15837356724*n^3 + 12355698460*n^4 - 6212542175*n^5 + 2144307578*n^6 - 526197678*n^7 + 93450369*n^8 - 12064836*n^9 + 1122618*n^10 - 73423*n^11 + 3206*n^12 - 84*n^13 + n^14.
a(n) = (n - 6)*(n - 5)*(n - 4)*(n - 3)*(n - 2)*(n - 1)*n*(n^7 - 63 n^6 + 1708 n^5 - 25795 n^4 + 234094 n^3 - 1275281 n^2 + 3858049 n - 4996032).
a(n) = Sum_{i=1..14} A334279(7,i)*n^i.
From Chai Wah Wu, Jan 19 2024: (Start)
a(n) = 15*a(n-1) - 105*a(n-2) + 455*a(n-3) - 1365*a(n-4) + 3003*a(n-5) - 5005*a(n-6) + 6435*a(n-7) - 6435*a(n-8) + 5005*a(n-9) - 3003*a(n-10) + 1365*a(n-11) - 455*a(n-12) + 105*a(n-13) - 15*a(n-14) + a(n-15) for n > 14.
G.f.: x^7*(-52370755920*x^7 - 27190754640*x^6 - 6557740560*x^5 - 959792400*x^4 - 92962800*x^3 - 6032880*x^2 - 246960*x - 5040)/(x - 1)^15. (End)

A334280 Irregular table read by rows: T(n, k) is the coefficient of x^k in the chromatic polynomial of the 1-skeleton of the n-dimensional demihypercube, 0 <= k <= 2^(n-1).

Original entry on oeis.org

0, 1, 0, -1, 1, 0, -6, 11, -6, 1, 0, -2790, 7467, -7852, 4300, -1346, 244, -24, 1, 0, -36586695600, 132041735820, -208601259660, 194243767689, -120509323400, 53195294240, -17371817260, 4296667608, -815202340, 119111090, -13339000, 1127752, -69860, 3000, -80, 1

Offset: 1

Peter Kagey, Apr 22 2020

			n\k| 0     1    2     3    4     5   6   7 8
---+----------------------------------------
  1| 0     1
  2| 0    -1    1
  3| 0    -6   11    -6    1
  4| 0 -2790 7467 -7852 4300 -1346 244 -24 1

Eric Weisstein's World of Mathematics, Chromatic Polynomial
Eric Weisstein's World of Mathematics, Halved Cube Graph
Wikipedia, Demihypercube

Cf. A048994 (simplex), A334278 (hypercube), A334279 (cross-polytope).

T[n_, k_] := Coefficient[ChromaticPolynomial[GraphData[{"HalvedCube", n}], x], x, k]

cp's OEIS Frontend

A334278 Irregular table read by rows: T(n, k) is the coefficient of x^k in the chromatic polynomial of the cubical graph Q_n, 0 <= k <= 2^n.

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A334281 Number of n-colorings of the vertices of the 4-dimensional cross polytope such that no two adjacent vertices have the same color.

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A342088 Triangle read by rows: T(n,k) is the number of n-colorings of the vertices of the k-dimensional cross polytope such that no two adjacent vertices have the same color. 0 <= k <= n.

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A342073 Number of n-colorings of the vertices of the 5-dimensional cross polytope such that no two adjacent vertices have the same color.

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A342074 Number of n-colorings of the vertices of the 6-dimensional cross polytope such that no two adjacent vertices have the same color.

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A342075 Number of n-colorings of the vertices of the 7-dimensional cross polytope such that no two adjacent vertices have the same color.

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A334280 Irregular table read by rows: T(n, k) is the coefficient of x^k in the chromatic polynomial of the 1-skeleton of the n-dimensional demihypercube, 0 <= k <= 2^(n-1).

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