Using Mathematica ( what else? ):
Pascal[n_]:=MatrixExp[Table[If[i == (j+1),j+1,0],{i,0,n},{j,0,n}]]
Usage: Pascal[n]//MatrixForm
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1 week ago
Open University pure maths study and research blog
Generating functions are one of the most surprising, useful, and clever inventions in discrete mathematics.
f(x+h)-f(x)
f'(x) = lim -----------
h-> 0 h
For example:
f(x) = x^n
f'(x) = n*x^(n-1)
a_n = {1, 16, 81, 256, ... }
f(n) = n^4
f'(n) = 1 + 4n + 6n^2 + 4n^3
f''(n) = 14 + 24n + 12n^2
f(3)(n) = 36 + 24n
f(4)(n) = 24
For an arbitrary sequence f:
f(m)(n) = Sum[(-1)^(k)*Binomial[m,k]*f[n-k+m],{k,0,m}]
Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before
(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)