a =>                 // a[] = coefficients in ascending order
a.map((_, k) =>      // for each coefficient at index k in a[]:
  a.reduce(          //   for each coefficient v at index n in a[],
    (t, v, n) =>     //   with t as the accumulator:
    t +              //     add to t ...
    v * (            //     ... v multiplied by S(n, k)
                     //     actually implemented as s(n) where s is ...
      s = n =>       //     ... a recursive function taking only n explicitly:
      n ?            //       if n is not 0:
        s(--n, k--)  //         compute S(n - 1, k - 1)
        + ++k * s(n) //         + k * S(n - 1, k)
      :              //       else:
        !k           //         return 1 if k = 0, or 0 otherwise
    )(n),            //     initial call to S
    0                //     start with t = 0
  )                  //   end of reduce()
)                    // end of map()

a =>                 // a[] = coefficients in ascending order
a.map((_, k) =>      // for each coefficient at index k in a[]:
  a.reduce(          //   for each coefficient v at index n in a[],
    (t, v, n) =>     //   with t as the accumulator:
    t +              //     add to t ...
    v * (            //     ... v multiplied by S(n, k)
                     //     actually implemented as s(n) where s is ...
      s = n =>       //     ... a recursive function taking only n explicitly:
      n ?            //       if n is not 0:
        s(--n, k--)  //         compute S(n - 1, k - 1)
        + ++k * s(n) //         + k * S(n - 1, k)
      :              //       else:
        !k           //         return 1 if k = 0, or 0 otherwise
    )(n),            //     initial call to s
    0                //     start with t = 0
  )                  //   end of reduce()
)                    // end of map()

JavaScript (ES6), 78 bytes

a=>a.map((_,k)=>a.reduce((t,v,n)=>t+v*(S=k=>n?--n+k*S(k)+S(k-1)-n++:!k)(k),0))

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a =>               // a[] = coefficients in ascending order
a.map((_, k) =>    // for each coefficient at index k in a[]:
  a.reduce(        //   for each coefficient v at index n in a[],
    (t, v, n) =>   //   with t as the accumulator:
    t +            //     add to t ...
    v * (          //     ... v multiplied by S(n, k)
                   //     actually implemented as S(k) where S is ...
      S = k =>     //     ... a recursive function taking only k explicitly:
      n ?          //       if n is not 0:
        --n        //         decrement n
        + k * S(k) //         compute k * S(k)
        + S(k - 1) //         + S(k - 1)
        - n++      //         restore n (and cancel out the earlier --n)
      :            //       else:
        !k         //         return 1 if k = 0, or 0 otherwise
    )(k),          //     initial call to S
    0              //     start with t = 0
  )                //   end of reduce()
)                  // end of map()

JavaScript (ES6), 77 bytes

a=>a.map((_,k)=>a.reduce((t,v,n)=>t+v*(s=n=>n?s(--n,k--)+ ++k*s(n):!k)(n),0))

Try it online!

a =>                 // a[] = coefficients in ascending order
a.map((_, k) =>      // for each coefficient at index k in a[]:
  a.reduce(          //   for each coefficient v at index n in a[],
    (t, v, n) =>     //   with t as the accumulator:
    t +              //     add to t ...
    v * (            //     ... v multiplied by S(n, k)
                     //     actually implemented as s(n) where s is ...
      s = n =>       //     ... a recursive function taking only n explicitly:
      n ?            //       if n is not 0:
        s(--n, k--)  //         compute S(n - 1, k - 1)
        + ++k * s(n) //         + k * S(n - 1, k)
      :              //       else:
        !k           //         return 1 if k = 0, or 0 otherwise
    )(n),            //     initial call to S
    0                //     start with t = 0
  )                  //   end of reduce()
)                    // end of map()

a =>               // a[] = coefficients in ascending order
a.map((_, k) =>    // for each coefficient at index k in a[]:
  a.reduce(        //   for each coefficient v at index n in a[],
    (t, v, n) =>   //   with t as the accumulator:
    t +            //     add to t ...
    v * (          //     ... v multiplied by S(n, k)
                   //     actually implemented as S(k) where S is ...
      S = k =>     //     ... a recursive function taking only k explicitly:
      n ?          //       if n is not 0:
        --n        //         decrement n
        + k * S(k) //         compute k * S(k)
        + S(k - 1) //         + S(k - 1)
        - n++      //         restore n (and cancels out the earlier --n)
      :            //       else:
        !k         //         return 1 if k = 0, or 0 otherwise
    )(k),          //     initial call to S
    0              //     start with t = 0
  )                //   end of reduce()
)                  // end of map()

a =>               // a[] = coefficients in ascending order
a.map((_, k) =>    // for each coefficient at index k in a[]:
  a.reduce(        //   for each coefficient v at index n in a[],
    (t, v, n) =>   //   with t as the accumulator:
    t +            //     add to t ...
    v * (          //     ... v multiplied by S(n, k)
                   //     actually implemented as S(k) where S is ...
      S = k =>     //     ... a recursive function taking only k explicitly:
      n ?          //       if n is not 0:
        --n        //         decrement n
        + k * S(k) //         compute k * S(k)
        + S(k - 1) //         + S(k - 1)
        - n++      //         restore n (and cancel out the earlier --n)
      :            //       else:
        !k         //         return 1 if k = 0, or 0 otherwise
    )(k),          //     initial call to S
    0              //     start with t = 0
  )                //   end of reduce()
)                  // end of map()