{"id":1250,"date":"2019-05-26T11:31:05","date_gmt":"2019-05-26T09:31:05","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=1250"},"modified":"2020-10-05T15:06:49","modified_gmt":"2020-10-05T13:06:49","slug":"triangle-de-pascal-construit-avec-python-et-latex","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2019\/05\/26\/triangle-de-pascal-construit-avec-python-et-latex\/","title":{"rendered":"Triangle de Pascal construit avec Python et LaTeX"},"content":{"rendered":"\n<p>Triangle de Pascal construit avec Python et LaTeX. Nous allons voir dans cet article comment construire le triangle de Pascal en \\(\\LaTeX\\) \u00e0 l&#8217;aide de Python.<\/p>\n\n\n\n<!--more-->\n\n\n\n<p>Avant de commencer, il est bon de savoir ce qu&#8217;est  le triangle de Pascal. Pour cela, vous pouvez consulter par exemple la page <a href=\"https:\/\/fr.wikipedia.org\/wiki\/Triangle_de_Pascal\" target=\"_blank\" rel=\"noreferrer noopener\">wikipedia<\/a>.<\/p>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_83 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2019\/05\/26\/triangle-de-pascal-construit-avec-python-et-latex\/#Triangle_de_Pascal_construit_avec_Python_et_LaTeX_le_code_Python\" >Triangle de Pascal construit avec Python et LaTeX: le code Python<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2019\/05\/26\/triangle-de-pascal-construit-avec-python-et-latex\/#Le_code_LaTeX\" >Le code \\(\\LaTeX\\)<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Triangle_de_Pascal_construit_avec_Python_et_LaTeX_le_code_Python\"><\/span>Triangle de Pascal construit avec Python et LaTeX: le code Python<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">def trianglePascal(n):\n    T = [[0] * (n+1) for p in range(n+1)]\n    for n in range(n+1):\n        if n == 0:\n            T[n][0] = 1\n        else:\n            for k in range(n+1):\n                if k == 0:\n                    T[n][0] = 1\n                else:\n                    T[n][k] = T[n-1][k-1] + T[n-1][k]\n    return T\n\nT = trianglePascal(9)<\/pre>\n\n\n\n<p>Ce premier code est int\u00e9ressant pour voir comment construire une matrice (dont les coefficients sont ceux du triangle de Pascal).<\/p>\n\n\n\n<p>On commence par initialiser notre matrice T en la remplissant de &#8220;0&#8221;. Puis on la remplit selon la propri\u00e9t\u00e9 bien connue : \\(\\binom{n}{k}=\\binom{n-1}{k-1}+\\binom{n-1}{k}\\).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_code_LaTeX\"><\/span>Le code \\(\\LaTeX\\)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>On va utiliser PythonTeX :<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"latex\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">% sous windows \n% pdflatex --shell-escape -synctex=1 -interaction=nonstopmode %.tex|python C:\\Users\\trash\\AppData\\Local\\Programs\\MiKTeX\\scripts\\pythontex\\pythontex.py %.tex|pdflatex --shell-escape -synctex=1 -interaction=nonstopmode %.tex\n% Sous Linux, remplacer \"--shell-espace\" par \"-write18\"\n\n\\documentclass[10pt,a0paper,landscape]{article}\n\\usepackage[utf8]{inputenc}\n\\usepackage[french]{babel}\n\\usepackage[T1]{fontenc}\n\\usepackage{lmodern}\n\\usepackage{diagbox}\n\\usepackage{pythontex}\n\\usepackage{nopageno}\n\\usepackage{colortbl}\n\\usepackage{tikz}\n\\usepackage[margin=5mm]{geometry}\n\\setlength{\\parindent}{0pt}\n\\newcommand{\\dashed}{\\tikz[baseline=1mm]\\draw[gray!50,dashed](0,0)--(0,0.5);}\n\\begin{document}\n\n\\begin{pycode}\nn = 50\n\ndef trianglePascal(n):\n    T = [[0] * (n+1) for p in range(n+1)]\n    for n in range(n+1):\n        if n == 0:\n            T[n][0] = 1\n        else:\n            for k in range(n+1):\n                if k == 0:\n                    T[n][0] = 1\n                else:\n                    T[n][k] = T[n-1][k-1] + T[n-1][k]\n    return T\n\n\nT = trianglePascal(n)\n\nprint('\\\\begin{tabular}{|>{\\\\columncolor{orange!10}}c|*{',n,'}{c!{\\\\dashed}}c|}\\\\rowcolor{orange!10}\\\\hline\\\\diagbox[height=8mm]{$n$}{$k$}')\nfor k in range(n+1):\n    print('&amp;',k)\n    \nfor j in range(n+1):\n\tprint('\\\\\\\\\\\\hline ',j)\n\tfor k in range(n+1):\n\t\tif k == 0:\n\t\t\tprint('&amp;1')\n\t\telse:\n\t\t\tif T[j][k] != 0:\n\t\t\t\tprint('&amp;',T[j][k])\n\t\t\telse:\n\t\t\t\tprint('&amp;')\n\nprint('\\\\\\\\\\\\hline\\\\end{tabular}')\n\\end{pycode} \n\n\\end{document}<\/pre>\n\n\n\n<p>Une pr\u00e9cision ici : j&#8217;ai souhait\u00e9 s\u00e9parer chaque colonne par des pointill\u00e9s. Pour cela, j&#8217;ai utilis\u00e9 une macro TiKZ (pour si peu, \u00e7a fait un peu mal quand-m\u00eame&#8230;) car le package <em>arydshln<\/em> (qui permet de le faire simplement) rentre en conflit avec <em>diagbox<\/em>&#8230; Il fallait donc en sacrifier un! (en fait, ils ne rentrent pas vraiment en conflit mais si on utilise des pointill\u00e9s dans le tableau, une bo\u00eete noire appara\u00eet \u00e0 la place de l&#8217;\u00e9tiquette &#8220;n&#8221;). On obtient le document suivant :<\/p>\n\n\n\n<div class=\"wp-block-file aligncenter\"><a  href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/05\/Triangle-de-Pascal.pdf\" data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">Triangle de Pascal<\/a><a  href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/05\/Triangle-de-Pascal.pdf\" class=\"wp-block-file__button\" download data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">T\u00e9l\u00e9charger<\/a><\/div>\n\n\n\n<p>Pour les curieux, voici un aper\u00e7u des premi\u00e8res lignes et colonnes:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/05\/Triangle-de-Pascal.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"143\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/05\/Triangle-de-Pascal-300x143.png\" alt=\"Triangle de Pascal construit avec Python et LaTeX\" class=\"wp-image-1257\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/05\/Triangle-de-Pascal-300x143.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/05\/Triangle-de-Pascal-600x286.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/05\/Triangle-de-Pascal-768x366.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/05\/Triangle-de-Pascal-1024x488.png 1024w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/05\/Triangle-de-Pascal.png 1510w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><figcaption>Aper\u00e7u du document affichant le triangle de Pascal pour <em>n<\/em> = 50<\/figcaption><\/figure><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Triangle de Pascal construit avec Python et LaTeX. Nous allons voir dans cet article comment construire le triangle de Pascal en \\(\\LaTeX\\) \u00e0 l&#8217;aide de Python.<\/p>\n","protected":false},"author":1,"featured_media":1257,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4,1,6,5],"tags":[111,65,110],"class_list":["post-1250","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-informatique","category-latex","category-mathematiques","category-python","tag-coefficients-binomiaux","tag-pythontex","tag-triangle-de-pascal"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.6 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Triangle de Pascal construit avec Python et LaTeX - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Nous allons voir dans cet article comment construire le fameux triangle de Pascal \u00e0 l&#039;aide de Python, et en LaTeX.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.mathweb.fr\/euclide\/2019\/05\/26\/triangle-de-pascal-construit-avec-python-et-latex\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Triangle de Pascal construit avec Python et LaTeX - 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