A137514 A triangular sequence from umbral calculus expansion of Simon Plouffe's rational polynomial for A002890: p(x,t) = exp(x*t)*(1 - 6*t + 9*t^2 - 4*t^3 + t^4)/(4*t - 1)/(2*t - 1).

1, 0, 1, 2, 0, 1, 12, 6, 0, 1, 120, 48, 12, 0, 1, 1680, 600, 120, 20, 0, 1, 31680, 10080, 1800, 240, 30, 0, 1, 766080, 221760, 35280, 4200, 420, 42, 0, 1, 22579200, 6128640, 887040, 94080, 8400, 672, 56, 0, 1, 778014720, 203212800, 27578880, 2661120, 211680, 15120, 1008, 72, 0, 1

Offset

1,4

Comments

Row sums:

{1, 1, 3, 19, 181, 2421, 43831, 1027783, 29698089, 1011695401, 39319102891}

The t's here are actually Sqrt[] of the variables that give Gamma(1,t) in the Hill reference and is the expansion of Plouffe's rational polynomial for A002890. So this result is related closely to Hill's Gamma(x,y) and seems to be a generalization of the A002890 polynomial.

References

Terrel L. Hill, Statistical Mechanics: Principles and Selected Applications, Dover, New York, 1956, page 336 ff

Links

Table of n, a(n) for n=1..55.

Formula

p(x,t) = exp(x*t)*(1 - 6*t + 9*t^2 - 4*t^3 + t^4)/(4*t - 1)/(2*t - 1) = Sum_{n>=0} P(x,n)*t^n/n!; out_n,m=n!*Coefficients(P(x,n)).

Example

Triangle begins:
  {1},
  {0, 1},
  {2, 0, 1},
  {12, 6, 0, 1},
  {120, 48, 12, 0, 1},
  {1680, 600, 120, 20, 0, 1},
  {31680, 10080, 1800, 240, 30, 0, 1},
  {766080, 221760, 35280, 4200, 420, 42, 0, 1},
  {22579200, 6128640, 887040, 94080, 8400, 672, 56, 0, 1},
  {778014720, 203212800, 27578880, 2661120, 211680, 15120, 1008, 72, 0, 1},
  ...

Mathematica

Clear[p, f, g] p[t_] = Exp[x*t]*(1 - 6*t + 9*t^2 - 4*t^3 + t^4)/(4*t - 1)/(2*t - 1); Table[ ExpandAll[n!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], {n, 0, 10}] a = Table[ CoefficientList[n!*SeriesCoefficient[; FullSimplify[Series[p[t], {t, 0, 30}]], n], x], {n, 0, 10}]; Flatten[a]

Crossrefs

Cf. A002890, A136264.

Sequence in context: A261161 A387648 A361951 * A367381 A322221 A328924

Adjacent sequences: A137511 A137512 A137513 * A137515 A137516 A137517

Keyword

nonn,uned,tabl

Author

Roger L. Bagula, Apr 23 2008

Status

approved