t: Weighted Taylor

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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                    NB. 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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