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A137943
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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)).
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0
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-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
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OFFSET
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1,4
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COMMENTS
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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.
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REFERENCES
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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
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LINKS
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FORMULA
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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)).
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EXAMPLE
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{-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}
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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