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 A137943 Triangle of coefficients associate with the expansion of the K_3 graph matric characteristic polynomial as a Sheffer sequence: M = {{0, 1, 1}, {1, 0, 1}, {1, 1, 0}} f(t)=-t^3+3t+2 p(x,t)=Exp[x,t)/(2*t^3+3*t^2-1)=exp(x*t)(t^3*f(1/t)). 0
 -1, 0, -1, -6, 0, -1, -12, -18, 0, -1, -216, -48, -36, 0, -1, -1440, -1080, -120, -60, 0, -1, -22320, -8640, -3240, -240, -90, 0, -1, -272160, -156240, -30240, -7560, -420, -126, 0, -1, -4717440, -2177280, -624960, -80640, -15120, -672, -168, 0, -1, -81285120, -42456960, -9797760, -1874880, -181440 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS The row sums are: {-1, -1, -7, -31, -301, -2701, -34531, -466747, -7616281, -135624601, -2728511551}; This sequence is a method of projecting the K_3 graph matrix on to a Sheffer sequence. REFERENCES Jonathan L. Gross and Thomas W. Tucker," Topological Graph Theory",Dover, New York,2001, page 10 figure 1.7 Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 149 LINKS FORMULA M = {{0, 1, 1}, {1, 0, 1}, {1, 1, 0}} f(t)=-t^3+3t+2 p(x,t)=Exp[x,t)/(2*t^3+3*t^2-1)=exp(x*t)(t^3*f(1/t))=Sum(P(x,n)*t^n/n!,{n,0,Infinity}) Out_n,m=n!*Coefficients(P(x,n)). EXAMPLE {-1}, {0, -1}, {-6, 0, -1}, {-12, -18, 0, -1}, {-216, -48, -36, 0, -1}, {-1440, -1080, -120, -60, 0, -1}, {-22320, -8640, -3240, -240, -90, 0, -1}, {-272160, -156240, -30240, -7560, -420, -126, 0, -1}, {-4717440, -2177280, -624960, -80640, -15120, -672, -168, 0, -1}, {-81285120, -42456960, -9797760, -1874880, -181440, -27216, -1008, -216, 0, -1}, {-1665619200, -812851200, -212284800, -32659200, -4687200, -362880, -45360, -1440, -270, 0, -1} MATHEMATICA Clear[p, b, a, x, y, t]; (*K_3 graph connection mathrix*) M = {{0, 1, 1}, {1, 0, 1}, {1, 1, 0}}; f[t_] = CharacteristicPolynomial[M, t]; p[t_] = ExpandAll[Exp[x*t]/(t^3*f[1/t])]; g = Table[ExpandAll[(n!)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[(n!)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a] CROSSREFS Cf. A000045. Sequence in context: A195445 A215080 A317446 * A202189 A202183 A227612 Adjacent sequences:  A137940 A137941 A137942 * A137944 A137945 A137946 KEYWORD tabl,uned,sign AUTHOR Roger L. Bagula, Apr 30 2008 STATUS approved

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Last modified September 21 00:17 EDT 2019. Contains 327252 sequences. (Running on oeis4.)