Riordan numbers

DefinitionRiordan numbers#✓fully proved

For a natural number \(n\), the \(n\)-th Riordan number is the integer

\[r_n = \sum_{k=0}^{n} (-1)^{n-k} \binom{n}{k} C_k,\]

where \(C_k\) is the \(k\)-th Catalan number.

def riordanNumber (n : Nat) : Int :=
  ∑ k ∈ Finset.range (n + 1),
    (-1 : Int) ^ (n - k) * ((n.choose k : Int) * ((catalan k : Int)))

Sources

• closed form (Kirgizov lines 436-438).

Used by 3

Used byRiordan numbers as an alternating Catalan sum · Log-convexity of the even-indexed Riordan numbers · Zero density of Riordan numbers modulo a prime

TheoremRiordan numbers as an alternating Catalan sum#✓fully proved

For every natural number \(n\), the \(n\)-th Riordan number \(r_n\) equals

\[\sum_{k=0}^{n} (-1)^{n-k} \binom{n}{k} C_k,\]

where \(C_k\) is the \(k\)-th Catalan number.

theorem riordanNumber_eq_sum (n : Nat) :
    riordanNumber n =
      ∑ k ∈ Finset.range (n + 1),
        (-1 : Int) ^ (n - k) * ((n.choose k : Int) * ((catalan k : Int)))

Uses 1

Statement usesRiordan numbers

2 fully proved

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