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 A137946 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. 0
 1, 0, 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 (list; table; graph; refs; listen; history; text; internal format)
 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 LINKS Table of n, a(n) for n=1..36. 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 Cf. A000045. Sequence in context: A028635 A028619 A062765 * A256856 A028603 A205966 Adjacent sequences: A137943 A137944 A137945 * A137947 A137948 A137949 KEYWORD nonn,tabl,uned AUTHOR Roger L. Bagula, Apr 30 2008 STATUS approved

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Last modified September 13 03:07 EDT 2024. Contains 375857 sequences. (Running on oeis4.)