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
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(P(x,n)*t^n/n!),{n,0,Infinity}]; out_n,m=n!*Coefficients(P(x,n)).
EXAMPLE
{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},
{30423859200, 7780147200, 1016064000, 91929600, 6652800, 423360, 25200, 1440, 90, 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
KEYWORD
AUTHOR
Roger L. Bagula, Apr 23 2008
STATUS
approved