OFFSET
1,4
COMMENTS
The row sums are:
{1, 0, 6, 12, 216, 1440, 22320, 272160, 4717440, 81285120, 1665619200}
This sequence is a method of projecting the K_3 graph matrix
on to a Sheffer sequence. This one is like that used to generate the Fibonacci numbers.
REFERENCES
Jonathan L. Gross and Thomas W. Tucker," Topologocal 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)=p(x,t)=1/(2*t^3+3*t^2-1)^x=1/(t^3*f(1/t))^x=Sum(P(x,n)*t^n/n!,{n,0,Infinity}) Out_n,m=n!(-1)^x*Coefficients(P(x,n)).
EXAMPLE
{1},
{},
{0, 6},
{0, 12},
{0, 108, 108},
{0, 720, 720},
{0, 7920, 11160, 3240},
{0, 90720, 136080, 45360},
{0, 1300320, 2222640, 1058400, 136080},
{0, 20563200, 37376640, 20079360, 3265920},
{0, 372314880, 726667200, 453146400, 106142400, 7348320}
MATHEMATICA
(*K_3 graph connection matrix*) M = {{0, 1, 1}, {1, 0, 1}, {1, 1, 0}}; f[t_] = CharacteristicPolynomial[M, t]; p[t_] = ExpandAll[1/(t^3*f[1/t])^x]; g = Table[ExpandAll[(n!*(-1)^x)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[(n!*(-1)^x)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10} Flatten[a]
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Apr 30 2008
STATUS
approved