bibkey: hanna2022a352700
authors: Paul D. Hanna
year: 2022
title: "OEIS A352700: Catalan inverse-power binomial series and modulo 2 and 3 congruences"
doi: null
url: https://oeis.org/A352700
claim: "Defines A(x) = sum_{n>=0} (binomial((n+1)(2n+1), n)/(2n+1)) x^n / C(x)^(n(2n+1)+1) over the Catalan series C(x) = 1 + x C(x)^2, and conjectures six modulo 2 and modulo 3 power-series and coefficient congruences (1)-(6)."
strata_touched:
  - D5/S3/Combinatorics/PowerSeries/HannaA352700CatalanCongruences
license: citation-only
triage: anchor

OEIS A352700

Verified locator

Source: https://oeis.org/A352700 (revision #7, March 31, 2022; authored by Paul D. Hanna, March 29, 2022).

The entry is unpaginated. Its OFFSET is 0,3, and its NAME defines the generating function:

G.f.: Sum_{n>=0} binomial((n+1)(2n+1),n)/(2n+1) * x^n / C(x)^(n(2n+1)+1), where C(x) = 1 + xC(x)^2 is the Catalan function (A000108).

Its COMMENTS state the six numbered conjectures:

Conjectures: given g.f. A(x), let C(x) = (1 - sqrt(1-4x))/(2x) be the Catalan power series (A000108), then (1) A(x)^3 = A(x^3) (mod 3), (2) A(x) = C(x) + x^2C(x)^3 (mod 3) = (2 - x)C(x) - 1 (mod 3), (3) A(x) = C(x) (mod 2), (4) a(n) = binomial(2n+1,n)/(2n+1) + 3binomial(2n-1,n-2)/(2n-1) (mod 3) for n >= 0, (5) a(n) = 2A000108(n) - A000108(n-1) (mod 3) for n >= 1, (6) a(n) = A000108(n) (mod 2) for n >= 0.

Its FORMULA field records the formal substitutions:

(1) A(x-x^2) = Sum_{n>=0} binomial((n+1)(2n+1),n)/(2n+1) * (x/(1-x))^n * (1-x)^((n+1)(2n+1)). (2) A(x/(1+x)^2) = Sum_{n>=0} binomial((n+1)(2n+1),n)/(2n+1) * x^n / (1+x)^((n+1)(2n+1)).

Its EXAMPLE field gives:

C(x) + x^2C(x)^3 = 1 + x + 3x^2 + 8x^3 + 23x^4 + 70x^5 + 222x^6 + 726x^7 + 2431x^8 + 8294x^9 + … + (C(2n,n)/(n+1) + C(2*n-1,n-2)3/(2n-1))*x^n + …

Its PROG (PARI) field computes a(n) by the finite degree-n truncation sum(m=0, n, binomial((m+1)*(2*m+1), m)/(2*m+1) * x^m / C^(m*(2*m+1)+1)).

For every natural m, 2*m + 1 divides binomial((m+1)*(2*m+1), m) in N, so each weight is an exact integer agreeing with rational division. Likewise 2*n + 1 divides binomial(2*n + 1, n) for every n >= 0, and 2*n - 1 divides 3 * binomial(2*n - 1, n - 2) for every n >= 2. In conjecture (4) and the EXAMPLE expansion of C(x) + x^2 * C(x)^3, the standard negative-lower-index binomial convention binomial(2*n - 1, n - 2) = 0 for n < 2 makes the second summand vanish at n = 0 and n = 1.

The companion comment "Odd terms seem to occur only at positions 2^n-1 for n >= 0" follows as a corollary of (6) together with the standard dyadic parity characterization of Catalan numbers, and is outside the six numbered conjectures (1)-(6) formalized as the primary target. Global literature completeness and continued external open status are ASSUMED-UNVERIFIED; no prior proof of (1)-(6) was found in the bounded OEIS and literature screen.

Primary locator: https://oeis.org/A352700/internal.