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FactorialSum

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Description

Summary

A polynomial times the factorial of the index, summed in closed form where the sum
telescopes: sum(k k!, k, 1, n) is (n + 1)! - 1, Sullivan and Mackey's Prob 2.7.17.

Remarks

P(k) k! is T(k + 1) - T(k) for T(k) = Q(k) k! exactly where
(k + 1) Q(k + 1) - Q(k) = P(k), and then the sum from a to b is
T(b + 1) - T(a). The equation is triangular in Q's coefficients from the top
down, the coefficient of k^(m + 1) fixing q_m, and its constant term is one
condition more: q_1 + q_2 + ... = p_0. Where that fails, no polynomial Q exists
and the sum is declined: sum(k!, k, 0, n), the left factorial, has no closed form in
factorials. This is Gosper's algorithm for the one family, where the certificate is a
polynomial.
 Over a >= 0, where a! is defined; a range that runs backwards is empty and
the sum zero, which is what the piecewise says, while the closed form at b = a - 1 is
zero already.
 https://github.com/asc-community/AngouriMath/issues/1409

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