t3_entries_derivatives
======================

.. py:function:: t3toolbox.backend.sampling_derivatives.t3_entries_derivatives(index, pp, x, order)

   .. code-block:: python

      def t3_entries_derivatives(
              index:  NDArray,                # int, shape=(d,)+W -- the grid points (one multi-index per W sample)
              pp:     typ.Sequence[NDArray],  # perturbation vectors P, len=d, elm_shape=W+(Ni,)
              x:      typ.Tuple[
                  typ.Sequence[NDArray],      # tucker_cores. len=d, elm_shape=C+(nUi,Ni)
                  typ.Sequence[NDArray],      # tt_cores.     len=d, elm_shape=C+(rLi,nUi,rR(i+1))
              ],                              # = TuckerTensorTrain.data
              order:  int,                    # highest derivative order
      ) -> NDArray:                           # entries-derivative jets, shape=(order+1,)+W+C


   Symmetric derivatives of an entry of a Tucker tensor train, in one repeated direction.

   The ``entries`` analog of :py:func:`t3_apply_derivatives` -- apply-derivatives with the up-index
   frame jet from slicing Tucker-core fibers at ``index`` (order 0) and contracting ``P`` (order 1).
   Returns ``y^(t) = d^t/ds^t apply(X, e_{index} + s P)|_0`` for ``t=0..order``: the Taylor data of
   the tensor's multilinear extension at grid corner ``index``, in direction ``P``. Index ``0`` is the
   ordinary entry ``X[index]``. Stacks ``order + W + C``. Verified vs :py:func:`dense_entries_derivatives`.