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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
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
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
KEYWORD
tabl,uned,sign
AUTHOR
Roger L. Bagula, Apr 30 2008
STATUS
approved