A138186 A triangular sequence based on expansion of the rational polynomial of A023054 as a Sheffer sequence: p(x,t)=Exp[x*t]*(1 - t^5)/((1 - t)*(1 - t^2)^2*(1 - t^3)).
1, 1, 1, 6, 2, 1, 24, 18, 3, 1, 168, 96, 36, 4, 1, 960, 840, 240, 60, 5, 1, 9360, 5760, 2520, 480, 90, 6, 1, 70560, 65520, 20160, 5880, 840, 126, 7, 1, 806400, 564480, 262080, 53760, 11760, 1344, 168, 8, 1, 7983360, 7257600, 2540160, 786240, 120960, 21168
Offset: 1
Examples
{1},
{1, 1},
{6, 2, 1},
{24, 18, 3, 1},
{168, 96, 36, 4, 1},
{960, 840, 240, 60, 5, 1},
{9360, 5760, 2520, 480, 90, 6, 1},
{70560, 65520, 20160, 5880, 840, 126, 7, 1},
{806400, 564480, 262080, 53760, 11760, 1344, 168, 8, 1},
{7983360, 7257600, 2540160, 786240, 120960, 21168, 2016, 216, 9, 1},
{105235200, 79833600, 36288000, 8467200, 1965600, 241920, 35280, 2880, 270, 10, 1}
Crossrefs
Cf. A023054.
Programs
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Mathematica
p[t_] = Exp[x*t]*(1 - t^5)/((1 - t)*(1 - t^2)^2*(1 - t^3)) Table[ ExpandAll[n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]
Formula
p(x,t)=Exp[x*t]*(1 - t^5)/((1 - t)*(1 - t^2)^2*(1 - t^3))=Sum(P(x,n)*t^n/n!,{n,0,Infinity}); Out_n,m=n!*Coefficients(P(x,n)).
Comments