(This article was originally published at Honglang Wang's Blog, and syndicated at StatsBlogs.)

Python is great and I think will be also great. For pure mathematics, it has lots of symbol calculations, since pure mathematics is abstract and powerful, like differential geometry, commutative algebra, algebraic geometry, and so on. However, science is nothing but experiment and computation. We also need powerful computational software to help us to carry out the result by powerful computation. Sage is your choice ! Since Sage claims that

Sageis a free open-source mathematics software system licensed under the GPL. It combines the power of many existing open-source packages into a common Python-based interface.Mission:Creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab.

Not only for pure mathematics, today I happened to see a blog post about using Sage to calculate high moments of Gaussian:

`var('m, s, t')`

`mgf(t) = exp(m*t + t^2*s^2/2)`

`for i in range(1, 11):`

`derivative(mgf, t, i).subs(t=0)`

`m`

`m^2 + s^2`

`m^3 + 3*m*s^2`

`m^4 + 6*m^2*s^2 + 3*s^4`

`m^5 + 10*m^3*s^2 + 15*m*s^4`

`m^6 + 15*m^4*s^2 + 45*m^2*s^4 + 15*s^6`

`m^7 + 21*m^5*s^2 + 105*m^3*s^4 + 105*m*s^6`

`m^8 + 28*m^6*s^2 + 210*m^4*s^4 + 420*m^2*s^6 + 105*s^8`

`m^9 + 36*m^7*s^2 + 378*m^5*s^4 + 1260*m^3*s^6 + 945*m*s^8`

`m^10 + 45*m^8*s^2 + 630*m^6*s^4 + 3150*m^4*s^6 + 4725*m^2*s^8 + 945*s^10`

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