%I #2 Mar 30 2012 17:34:26
%S 1,0,0,6,0,12,0,108,108,0,720,720,0,7920,11160,3240,0,90720,136080,
%T 45360,0,1300320,2222640,1058400,136080,0,20563200,37376640,20079360,
%U 3265920,0,372314880,726667200,453146400,106142400,7348320
%N 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)=1/(2*t^3+3*t^2-1)^x=1/(t^3*f(1/t))^x.
%C The row sums are:
%C {1, 0, 6, 12, 216, 1440, 22320, 272160, 4717440, 81285120, 1665619200}
%C This sequence is a method of projecting the K_3 graph matrix
%C on to a Sheffer sequence. This one is like that used to generate the Fibonacci numbers.
%D Jonathan L. Gross and Thomas W. Tucker," Topologocal Graph Theory",Dover, New York,2001, page 10 figure 1.7
%D Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 149
%F 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)).
%e {1},
%e {},
%e {0, 6},
%e {0, 12},
%e {0, 108, 108},
%e {0, 720, 720},
%e {0, 7920, 11160, 3240},
%e {0, 90720, 136080, 45360},
%e {0, 1300320, 2222640, 1058400, 136080},
%e {0, 20563200, 37376640, 20079360, 3265920},
%e {0, 372314880, 726667200, 453146400, 106142400, 7348320}
%t (*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]
%Y Cf. A000045.
%K nonn,tabl,uned
%O 1,4
%A _Roger L. Bagula_, Apr 30 2008