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A method for deterining the mod-$3^k$ behaviour of recursive sequences

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Christian Krattenthaler (Universität Wien), Thomas W. Müller (Queen Mary)

Combinatorics · Submitted 13 Aug 2013

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This leads to numerous new results, including many extensions of known results to higher powers of 3.

how it is stated · asserted 1.00 read the record →

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arXiv ID1308.2856 (version 1)
Pith IDV2HZT47X, Pith's permanent record of this paper and its review
CitationChristian Krattenthaler (Universit\"at Wien), Thomas W. M\"uller (Queen Mary). "A method for deterining the mod-$3^k$ behaviour of recursive sequences." arXiv:1308.2856 (2013). https://pith.science/paper/V2HZT47X
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The compiled record

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,990 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.42 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We present a method for obtaining congruences modulo powers of 3 for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Motzkin numbers, Riordan numbers, Schr\"oder numbers, Eulerian numbers, trinomial coefficients, Delannoy numbers, and to functions counting free subgroups of finite index in the inhomogeneous modular group and its lifts. This leads to numerous new results, including many extensions of known results to higher powers of 3.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We present a method for obtaining congruences modulo powers of 3 for sequences given by recurrences of finite depth with polynomial coefficients. contribution 0.75 0.94 0.36 asserted 1.00
s1 We apply this method to Catalan numbers, Motzkin numbers, Riordan numbers, Schr\"oder numbers, Eulerian numbers, trinomial coefficients, Delannoy numbers, and to functions counting free subgroups of finite index in the inhomogeneous modular group and its lifts. contribution 0.85 0.92 0.09 asserted 1.00
s2 This leads to numerous new results, including many extensions of known results to higher powers of 3. result 0.86 0.90 0.45 asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from proof or derivation 1.00.

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  • figures and tables
  • references and citing sentences
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  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We present a method for obtaining congruences modulo powers of 3 for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Motzkin numbers, Riordan numbers, Schr\"oder numbers, Eulerian numbers, trinomial coefficients, Delannoy numbers, and to functions counting free subgroups of finite index in the inhomogeneous modular group and its lifts. This leads to numerous new results, including many extensions of known results to higher powers of 3.

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References
How the literature uses it

7 arXiv papers cite this in 14 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 12 sentences without a confident label

Uses its method or tool 1 paper
Cites it as background 1 paper
  • Kauers, Krattenthaler, and Müller developed the first systematic methods for producing congruences modulo $2^α$ in [ref] and modulo $3^α$ in [ref] for a large family of differentially algebraic sequences, including many algebraic sequences.

    Automatic congruences for diagonals of rational functions · Overview · 0.83

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