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.

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

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

Triangle begins:
  {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,tabf,uned

Author

Roger L. Bagula, Apr 30 2008

Status

approved