|
Search: id:A137514
|
|
|
| A137514 |
|
A triangular sequence from Umbral Calculus expansion of S. 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
|
|
| 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
(list; table; graph; listen)
|
|
|
OFFSET
|
1,4
|
|
|
COMMENT
|
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 the 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 Selcted 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
|
Cf. A002890, A136264.
Sequence in context: A072551 A091803 A123002 this_sequence A069845 A091397 A119818
Adjacent sequences: A137511 A137512 A137513 this_sequence A137515 A137516 A137517
|
|
KEYWORD
|
nonn,uned,tabl
|
|
AUTHOR
|
Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 23 2008
|
|
|
Search completed in 0.002 seconds
|
