Morgan–Voyce polynomials

DefinitionMorgan–Voyce coefficients#✓fully proved

For natural numbers \(n\) and \(k\), the Morgan–Voyce coefficient \(d(n, k)\) is the binomial coefficient \(\binom{n + k}{n - k}\) if \(k \le n\), and \(0\) if \(k > n\).

These numbers form the lower-triangular Riordan array \(\left(\frac{1}{1 - x}, \frac{x}{(1 - x)^2}\right)\), OEIS A085478.

def morganVoyceCoeff (n k : ℕ) : ℕ :=
  if k ≤ n then (n + k).choose (n - k) else 0

Used by 1

Used byMorgan–Voyce polynomials

DefinitionMorgan–Voyce polynomials#✓fully proved

For a semiring \(R\) and a natural number \(n\), the Morgan–Voyce polynomial of index \(n\) over \(R\) is

\[b_n(x) = \sum_{k=0}^{n} d(n, k)\, x^k \in R[x],\]

where \(d(n, k) = \binom{n + k}{n - k}\) is the Morgan–Voyce coefficient, read in \(R\) through the map from the natural numbers.

The coefficients of \(b_n\) form row \(n\) of the Riordan array \(\left(\frac{1}{1 - x}, \frac{x}{(1 - x)^2}\right)\), OEIS A085478.

noncomputable def morganVoycePolynomial (R : Type*) [Semiring R] (n : ℕ) :
    Polynomial R :=
  Finset.sum (Finset.range (n + 1)) fun k =>
    Polynomial.C ((morganVoyceCoeff n k : ℕ) : R) * Polynomial.X ^ k

Sources

• OEIS A085478.

Uses 1

Statement usesMorgan–Voyce coefficients

2 fully proved

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