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3

The expressions for the coefficients of the Pochhammer symbol are in fact well-known (see e.g. Concrete Mathematics): $$\prod_{k=0}^{n-1}(x+k)=\sum_{k=0}^n \left[{n}\atop{k}\right]x^k$$ where $\left[{n}\atop{k}\right]$ is a Stirling cycle number. In Mathematica, this corresponds to (-1)^(n - k) StirlingS1[n, k]. Table[Product[x + k, {k, 0, n - 1}] == ...


4

expr = (-1 + (E^(w[1, 0] - 1.55432 w[1, 1]) - E^(-w[1, 0] + 1.55432 w[1, 1]))^2 / (E^(w[1, 0] - 1.55432 w[1, 1]) + E^(-w[1, 0] + 1.55432 w[1, 1]))^2) expr /. Power[E, Plus[x_, y__]] :> Inactive[Times][Power[E, x], Power[E, y]]/. {Power[E, Times[c_, w[i_, j_], ___]] :> Power[T[i, j], Sign[c]], Power[E, w[i_, j_]] ...


1

f = Expand[# Denominator@FunctionExpand@FactorialPower[#, -#2 + 1]] &; f[t, 3] Grid[{#, f[x, #], Rest@CoefficientList[f[x, #], x]} & /@ Range[5], Alignment -> Left] Or f2 = Expand[FunctionExpand[FactorialPower[#, #2]] /. i_Integer :> -i] &; Grid[{#, f2[x, #]} & /@ Range[5], Alignment -> Left]


3

f[n_, x_] := Expand[Fold[(#2 + x - 2) #1 & , Range[n + 1]]] e.g. Grid[{f[#, t], CoefficientList[f[#, t], t]} & /@ Range[0, 10]] Just to illustrate unexpanded polynomial: g[n_, x_] := Fold[(#2 + x - 2) #1 & , Range[n + 1]] TableForm[{#, g[#, u], g[#, 1], #!} & /@ Range[0, 5], TableHeadings -> {None, {"n", "p(u)", "p(1)", "n!"}}] ...



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