Sum of kⁿ/k! and kⁿ·rᵏ/k! over an infinite range is unevaluated (Dobiński's formula) #576

Environment: @cortex-js/compute-engine 0.187.1 (main)

ce.box(['Sum', ['Divide', ['Power', 'k', 3], ['Factorial', 'k']], ['Limits', 'k', 1, 'PositiveInfinity']]).evaluate();
// Sum(k^3 / k!, k = 1..∞)        (simplify() the same; .N() is the float 13.591409142296198)

ce.parse('\\sum_{k=0}^\\infty \\frac{k^2 x^k}{k!}').evaluate();   // unevaluated
ce.parse('\\sum_{k=0}^\\infty \\frac{k^2 + k}{k!}').evaluate();   // unevaluated

Σ_{k≥0} 1/k! is e and Σ_{k≥a} xᵏ/k! is eˣ − Σ_{k<a} xᵏ/k! (namedSeriesClosedForm), but any power of k over k! is not recognized.

Wolfram gives Sum[k^3/k!, {k, 1, Infinity}] = 5 E, Sum[k^2 x^k/k!, {k, 0, Infinity}] = E^x (x + x^2), Sum[(k^2 + k)/k!, {k, 0, Infinity}] = 3 E and Sum[k^2/k!, {k, 2, Infinity}] = -1 + 2 E.

The values are Dobiński's formula Σ_{k≥0} kⁿ/k! = e·Bₙ (Bₙ the Bell number) and its generalization with a ratio, Σ kⁿ·rᵏ/k! = eʳ·Tₙ(r), Tₙ(r) = Σⱼ S(n, j)·rʲ the Touchard polynomial (Stirling numbers of the second kind); Σ k³/k! = e·B₃ = 5e. The series are entire in r, so the closed form has no convergence condition, as the existing rᵏ/k! entry.

A PR is ready.