A137514 - OEIS

%I #6 Apr 26 2013 22:43:00

%S 1,0,1,2,0,1,12,6,0,1,120,48,12,0,1,1680,600,120,20,0,1,31680,10080,

%T 1800,240,30,0,1,766080,221760,35280,4200,420,42,0,1,22579200,6128640,

%U 887040,94080,8400,672,56,0,1,778014720,203212800,27578880,2661120

%N 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).

%C Row sums:

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

%C The t's here are actually Sqrt[] of the variables that give Gamma(1,t)

%C in the Hill reference and is the expansion of Plouffe's

%C rational polynomial for A002890. So this result is related closely

%C to Hill's Gamma(x,y) and seems to be a generalization of the A002890 polynomial.

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

%F 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)).

%e {1},

%e {0, 1},

%e {2, 0, 1},

%e {12, 6, 0, 1},

%e {120, 48, 12, 0, 1},

%e {1680, 600, 120, 20, 0, 1},

%e {31680, 10080, 1800, 240, 30, 0, 1},

%e {766080, 221760, 35280, 4200, 420, 42, 0, 1},

%e {22579200, 6128640, 887040, 94080, 8400, 672, 56, 0, 1},

%e {778014720, 203212800, 27578880, 2661120, 211680, 15120, 1008, 72, 0, 1},

%e {30423859200, 7780147200, 1016064000, 91929600, 6652800, 423360, 25200, 1440, 90, 0, 1}

%t 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]

%Y Cf. A002890, A136264.

%K nonn,uned,tabl

%O 1,4

%A _Roger L. Bagula_, Apr 23 2008