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A182793 Number of n-colorings of the Triangle Graph of order 8. 12
0, 0, 0, 6, 1031276544, 4826149802070660, 316827094291524894720, 1595091571660292411606250, 1592275064882420035249606656, 526249245643156296389047576104, 78022473527414400196098852126720, 6300701001267935948773824927446190 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The Triangle Graph of order 8 has 8 rows with k vertices in row k. 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 36 vertices and 84 edges altogether.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Triangle Graph

Wikipedia, Chromatic polynomial

FORMULA

a(n) = n^36 -84*n^35 + ... (see Maple program).

a(n) = (n^30 + ... )*n*(n-1)*(n-2)^4 (see PARI program), therefore all terms are divisible by 6. - M. F. Hasler, Dec 02 2010

MAPLE

a:= n-> n^36 -84*n^35 +3437*n^34 -91266*n^33 +1767948*n^32 -26626641*n^31 +324474230*n^30 -3287527515*n^29 +28241112564*n^28 -208720581316*n^27 +1342098781876*n^26 -7574085510428*n^25 +37773151152128*n^24 -167375021582772*n^23 +661739022592885*n^22 -2341944556478962*n^21 +7436934470326959*n^20 -21224613967949058*n^19 +54488667645973816*n^18 -125859887740997948*n^17 +261444368727996373*n^16 -487829426279117443*n^15 +816027319948726718*n^14 -1220298815193350831*n^13 +1625157969312740380*n^12 -1917859440184087949*n^11 +1992559474100473934*n^10 -1807335902805940076*n^9 +1415695106519940144*n^8 -943996557462968752*n^7 +525570615466126368*n^6 -237792323595423264*n^5 +84014216771282688*n^4 -21747100909979904*n^3 +3668087119290368*n^2 -302469084548608*n: seq(a(n), n=0..12);

PROG

(PARI) a(n) = n*(n-1)*(n-2)^4*(n^30 -15*(5*n^20 -182*n^19 -73212*n^17 +968723*n^16 -10321679*n^15 +90965902*n^14 -42239514291692*n^5 +728948069669224)*n^9 -64240*n^27 +10138842074*n^22 -64422107890*n^21 +353781404418*n^20 -1692797609642*n^19 +7100833446102*n^18 -26231755759998*n^17 +85617623199383*n^16 -247408302649363*n^15 -1437889343008038*n^13 +2888477744794634*n^12 -5124456558208194*n^11 +8000185529836163*n^10 +12990665090694358*n^8 -13287807554341505*n^7 +11549829535832291*n^6 -8378308904565234*n^5 +4943464695686292*n^4 -2282977532565696*n^3 +775401219820384*n^2 -172542491602784*n +18904317784288) \\ - M. F. Hasler, Dec 02 2010

CROSSREFS

8th column of A182797. Cf. A178435, A182798, A182788, A182789, A182790, A182791, A182792, A182794, A182795, A182796.

Sequence in context: A337083 A193150 A307896 * A058431 A045518 A058457

Adjacent sequences:  A182790 A182791 A182792 * A182794 A182795 A182796

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Dec 02 2010

STATUS

approved

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Last modified October 21 23:34 EDT 2021. Contains 348160 sequences. (Running on oeis4.)