These are not B-splines:
These are called discrete B-splines. They span a function spaced called discrete splines, which are analogous to splines, but defined in terms of a suitable discrete notion of smoothness.
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Discrete splines have continuous discrete derivatives at their knots (rather than continuous derivatives, as splines do).
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They have important computational properties, like the fact that interpolation within the space of discrete splines can be done in constant-time.
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They are intimately connected to trend filtering (they provide the basis representation that underlies the trend filtering estimator).
For more background, see the monograph: "Divided differences, falling factorials, and discrete splines: Another look at trend filtering and related problems".
The dspline package provides tools for computations with discrete splines. The
core routines are written in C++ for efficiency. See the
reference index for a
summary of the tools that are available.