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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).
0
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
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
Sequence in context: A123002 A261161 A361951 * A367381 A322221 A328924
KEYWORD
nonn,uned,tabl
AUTHOR
Roger L. Bagula, Apr 23 2008
STATUS
approved