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Search: id:A137946
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| A137946 |
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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)=1/(2*t^3+3*t^2-1)^x=1/(t^3*f(1/t))^x. |
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| 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; listen)
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OFFSET
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1,4
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COMMENT
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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.
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REFERENCES
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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
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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)=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)).
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EXAMPLE
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{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}
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MATHEMATICA
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(*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]
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CROSSREFS
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Cf. A000045.
Sequence in context: A028635 A028619 A062765 this_sequence A028603 A069828 A057401
Adjacent sequences: A137943 A137944 A137945 this_sequence A137947 A137948 A137949
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KEYWORD
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nonn,tabl,uned
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 30 2008
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