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

Author

Roger L. Bagula, Apr 23 2008

Keywords

• nonn
• uned
• tabl

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.

Examples

			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},
  ...
		

References

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

Crossrefs

Cf. A002890, A136264.

Programs

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]

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)).