A091940 -id:A091940 - cp's OEIS frontend

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.

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!).

A123659 a(n) = 1 + n^4 + n^6 + n^9 + n^10 + n^14.

Original entry on oeis.org

6, 18001, 4862512, 269750529, 6115250626, 78434755921, 678546021756, 4399254736897, 22880667197854, 100011001010001, 379778130741736, 1283985544700161, 3937524853545882, 11112316748827729, 29193541130581876

Offset: 1

Jonathan Vos Post, Oct 05 2006

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001358, A002523, A091940, A123177, A123656-A123659.

[1 + n^4 + n^6 + n^9 + n^10 + n^14: n in [1..25]]; // G. C. Greubel, Oct 17 2017

Table[1 + n^4 + n^6 + n^9 + n^10 + n^14, {n, 1, 50}] (* G. C. Greubel, Oct 17 2017 *)

for(n=1,25, print1(1 + n^4 + n^6 + n^9 + n^10 + n^14, ", ")) \\ G. C. Greubel, Oct 17 2017

a(n) = 1 + n^4 + n^6 + n^9 + n^10 + n^14.

A131472 a(n) = n^6 + n.

Original entry on oeis.org

0, 2, 66, 732, 4100, 15630, 46662, 117656, 262152, 531450, 1000010, 1771572, 2985996, 4826822, 7529550, 11390640, 16777232, 24137586, 34012242, 47045900, 64000020, 85766142, 113379926, 148035912, 191103000, 244140650, 308915802

Offset: 0

Mohammad K. Azarian, Jul 27 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A002378, A007531, A034262, A091940, A058895, A059826.

[n^6+n: n in [0..30]]; // _Vincenzo Librandi+, Oct 01 2011

Table[n^6+n,{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, May 12 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,2,66,732,4100,15630,46662},60] (* Harvey P. Dale, May 03 2012 *)

G.f.: 2*x*(1 + 26*x + 156*x^2 + 146*x^3 + 31*x^4)/(1 - x)^7. - R. J. Mathar, Nov 14 2007
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7); a(0)=0, a(1)=2, a(2)=66, a(3)=732, a(4)=4100, a(5)=15630, a(6)=46662. - Harvey P. Dale, May 03 2012
E.g.f.: exp(x)*x*(2 + 31*x + 90*x^2 + 65*x^3 + 15*x^4 + x^5). - Stefano Spezia, Oct 08 2022

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)

A380589 Number of n-colorings of the Hypercube Graph Q5.

Original entry on oeis.org

0, 0, 2, 1185282, 130253748108, 2157531034816940, 7905235551766437150, 7365707045872206479742, 2337101560809838105414712, 327425229254999498091796728, 24489214732779742874109277530, 1119349138930999380736025706650, 34471067091433681765512048700932

Offset: 0

Alois P. Heinz, Jan 27 2025

The Hypercube Graph Q5 has 32 vertices and 80 edges.

All terms are even.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Hypercube Graph
Eric Weisstein's World of Mathematics, Chromatic Polynomial
Index entries for linear recurrences with constant coefficients, signature (33, -528, 5456, -40920, 237336, -1107568, 4272048, -13884156, 38567100, -92561040, 193536720, -354817320, 573166440, -818809200, 1037158320, -1166803110, 1166803110, -1037158320, 818809200, -573166440, 354817320, -193536720, 92561040, -38567100, 13884156, -4272048, 1107568, -237336, 40920, -5456, 528, -33, 1).

Column k=5 of A342128.

Cf. A001477, A002378, A091940, A140986, A158348, A334278.

a:= n-> (((((((((((((((((((((((((((((((n-80)*n+3160)*n-82080)*n+1575420)*n
    -23805776)*n+294640000)*n-3068289720)*n+27406254870)*n-212981036784)*n
    +1455643449120)*n-8822129447280)*n+47712047044920)*n-231347639674200)*n
    +1009138022379076)*n-3968583456247214)*n+14086095737441185)*n-45124968898112160)*n
    +130327084318442384)*n-338572422663483544)*n+788328935798745052)*n
    -1636781898149840504)*n+3009654466362869780)*n-4856773984500880124)*n
    +6797172300402030636)*n-8122089299204814072)*n+8114599308192145448)*n
    -6584797184952049568)*n+4160914137061367054)*n-1915734714629493936)*n
    +569711421560808713)*n-81768640551939777)*n:
seq(a(n), n=0..12);

a(n) = n^32 - 80*n^31 + 3160*n^30 - ... (see Maple program).

A123657 a(n) = 1 + n^4 + n^6 + n^9.

Original entry on oeis.org

4, 593, 20494, 266497, 1969376, 10125649, 40473658, 134483969, 387958492, 1001010001, 2359733894, 5162787073, 10609354744, 20668614737, 38454800626, 68736319489, 118612097588, 198393407569, 322734873982, 512064160001

Offset: 1

Jonathan Vos Post, Oct 04 2006

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A001358, A002523, A091940, A123177, A123656-A123659.

[1 + n^4 + n^6 + n^9: n in [1..25]]; // G. C. Greubel, Oct 17 2017

Table[1 + n^4 + n^6 + n^9, {n, 1, 50}] (* G. C. Greubel, Oct 17 2017 *)

a(n)=n^9+n^6+n^4+1 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = 1 + n^4 + n^6 + n^9 = 1001010001 (base n).

G.f.: -x*(x^9 -8*x^8 -406*x^7 -14592*x^6 -88496*x^5 -156316*x^4 -87762*x^3 -14744*x^2 -553*x -4)/(x-1)^10. - Colin Barker, May 27 2012

A123665 a(n) = Sum_{k=1..21} n^A001358(k).

Original entry on oeis.org

22, 471260364628084305, 6457022669043550542502557676, 105149403852520725445003265581519105, 41911381174488637014293971538580334000626

Offset: 1

Jonathan Vos Post, Oct 04 2006

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001358, A002523, A091940, A123177, A123656-A123665.

[1 + n^4 + n^6 + n^9 + n^10 + n^14 + n^15 + n^21 + n^22 + n^25 +
n^26 + n^33 + n^34 + n^35 + n^38 + n^39 + n^46 + n^49 + n^51 + n^55 + n^57 + n^58: n in [1..50]]; // G. C. Greubel, Oct 26 2017

Table[1 + n^4 + n^6 + n^9 + n^10 + n^14 + n^15 + n^21 + n^22 + n^25 +
  n^26 + n^33 + n^34 + n^35 + n^38 + n^39 + n^46 + n^49 + n^51 +
  n^55 + n^57 + n^58, {n, 1, 50}] (* G. C. Greubel, Oct 26 2017 *)

for(n=1,50, print1(1 + n^4 + n^6 + n^9 + n^10 + n^14 + n^15 + n^21 + n^22 + n^25 + n^26 + n^33 + n^34 + n^35 + n^38 + n^39 + n^46 + n^49 + n^51 + n^55 + n^57 + n^58, ", ")) \\ G. C. Greubel, Oct 26 2017

a(n) = 1 +n^4 +n^6 +n^9 +n^10 +n^14 +n^15 +n^21 +n^22 +n^25 +n^26 + n^33 +n^34 +n^35 +n^38 +n^39 +n^46 +n^49 +n^51 +n^55 +n^57 +n^58.

Better name from Joerg Arndt, May 23 2021

A190578 a(n) = n^7 + n.

Original entry on oeis.org

0, 2, 130, 2190, 16388, 78130, 279942, 823550, 2097160, 4782978, 10000010, 19487182, 35831820, 62748530, 105413518, 170859390, 268435472, 410338690, 612220050, 893871758, 1280000020, 1801088562, 2494357910, 3404825470, 4586471448

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 12 2011

a(n) = n^7 + n, A005843 for k=1, A002378 for k=2, A034262 for k=3, A091940 for k=4, A131471 for k=5, A131472 for k=6.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A005843, A002378, A034262, A091940, A131471, A131472.

[n^7+n: n in [0..30]]; // Vincenzo Librandi, Sep 30 2011

Mathematica
```
k=7; Table[n^k+n,{n,0,50}]
```

Offset changed from 1 to 0 by Vincenzo Librandi, Sep 30 2011

cp's OEIS Frontend

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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A123659 a(n) = 1 + n^4 + n^6 + n^9 + n^10 + n^14.

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Magma

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A131472 a(n) = n^6 + 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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A380589 Number of n-colorings of the Hypercube Graph Q5.

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Maple

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A123657 a(n) = 1 + n^4 + n^6 + n^9.

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