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Weighted Taylor u t:  0 0 0  

The result of u t: k is (!k)*u t. k . In other words, the coefficients produced by t: are the Taylor coefficients weighted by the factorial. As a consequence, the coefficients produced by it when applied to functions of the exponential family show simple patterns. For this reason it is sometimes called the exponential generating function.
 

For example:
   k=: i. 12
   ^ t: k
1 1 1 1 1 1 1 1 1 1 1 1

   %@^t: k                    Decaying exponential
1 _1 1 _1 1 _1 1 _1 1 _1 1 _1

   sin =: 1&o. [. cos =: 2&o.
   sinh=: 5&o. [. cosh=: 6&o.
   exp=: ^ [. dec=: %@^

   (exp t:,dec t:,sinh t:,cosh t:,sin t:,:cos t:) k
1  1  1  1 1  1  1  1 1  1  1  1
1 _1  1 _1 1 _1  1 _1 1 _1  1 _1
0  1  0  1 0  1  0  1 0  1  0  1
1  0  1  0 1  0  1  0 1  0  1  0
0  1  0 _1 0  1  0 _1 0  1  0 _1
1  0 _1  0 1  0 _1  0 1  0 _1  0


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