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On the expansion of the power of any polynomial 1+x+x^2+x^3+x^4+etc

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Leonhard Euler

History and Overview · Submitted 20 May 2005

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

The authors' main claim In this paper, Euler gives the general trionomial coefficient as a sum of the binomial coefficients, the general quadrinomial coefficient as a sum of the binomial and trinomial coefficients, the general quintonomial coefficient as a sum of the binomial and quadrinomial coefficients, and gives a general determination of the coefficients of the expansion of any polynomial (1+x+x^2+...+x^m)^n as a sum of the coefficients of lower degree polynomial coefficients.

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arXiv IDmath/0505425 (version 1)
Pith IDCDMK6HQQ, Pith's permanent record of this paper and its review
CitationLeonhard Euler. "On the expansion of the power of any polynomial 1+x+x^2+x^3+x^4+etc." arXiv:math/0505425 (2005). https://pith.science/paper/CDMK6HQQ
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 1,748 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.05 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.13 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

In this paper, Euler gives the general trionomial coefficient as a sum of the binomial coefficients, the general quadrinomial coefficient as a sum of the binomial and trinomial coefficients, the general quintonomial coefficient as a sum of the binomial and quadrinomial coefficients, and gives a general determination of the coefficients of the expansion of any polynomial (1+x+x^2+...+x^m)^n as a sum of the coefficients of lower degree polynomial coefficients.

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 In this paper, Euler gives the general trionomial coefficient as a sum of the binomial coefficients, the general quadrinomial coefficient as a sum of the binomial and trinomial coefficients, the general quintonomial coefficient as a sum of the binomial and quadrinomial coefficients, and gives a general determination of the coefficients of the expansion of any polynomial (1+x+x^2+...+x^m)^n as a sum of the coefficients of lower degree polynomial coefficients. contribution 0.81 0.75 0.88 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.99.

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  • P abstract
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  • S sections
  • T body sentences
  • C citations
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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

In this paper, Euler gives the general trionomial coefficient as a sum of the binomial coefficients, the general quadrinomial coefficient as a sum of the binomial and trinomial coefficients, the general quintonomial coefficient as a sum of the binomial and quadrinomial coefficients, and gives a general determination of the coefficients of the expansion of any polynomial (1+x+x^2+...+x^m)^n as a sum of the coefficients of lower degree polynomial coefficients.

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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

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