{"id":61927,"date":"2021-05-29T09:38:45","date_gmt":"2021-05-29T07:38:45","guid":{"rendered":"https:\/\/wpcalc.com\/?p=61927"},"modified":"2026-09-30T20:37:38","modified_gmt":"2026-09-30T17:37:38","slug":"leibniz-harmonic-triangle-generator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/leibniz-harmonic-triangle-generator\/","title":{"rendered":"Leibniz Harmonic Triangle Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Generate Leibniz Triangle Values<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"61927\" data-favorite-title=\"Leibniz Harmonic Triangle\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/leibniz-harmonic-triangle-generator\/\" data-favorite-excerpt=\"Generate the Leibniz Harmonic Triangle for up to 150 rows, with every entry shown as an exact fraction in lowest terms. Just enter the number of rows and read off the values, each computed as (1\/k) \u00d7 C(n,k).\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-arithmetic\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/mathematics\/leibniz-harmonic-triangle-generator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-61927\" data-post=\"61927\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Number of rows<\/label><input type=\"text\" name=\"number_of_rows\" id=\"field-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"5\" min=\"1\" max=\"150\" value=\"5\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"61927\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-value nowrap\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"61927\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Leibniz Harmonic Triangle Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ a_{n,k} = \\frac{1}{k}\\binom{n}{k} = \\frac{n!}{k \\cdot k!\\,(n-k)!} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(n\\) \u2014 row number, counted from 1<\/li>\n<li>\\(k\\) \u2014 position of the entry within the row, \\(1 \\le k \\le n\\)<\/li>\n<li>\\(\\binom{n}{k}\\) \u2014 the binomial coefficient, read as \"\\(n\\) choose \\(k\\)\"<\/li>\n<\/ul>\n<p>Each entry of the triangle is the binomial coefficient from Pascal's triangle divided by its position \\(k\\) within the row, then reduced to lowest terms.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Leibniz Harmonic Triangle \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> generate the triangle with 5 rows (the calculator's default value). To find the entry in row \\(n = 5\\) at position \\(k = 3\\):<\/p>\n<p>\\[ a_{5,3} = \\frac{1}{3}\\binom{5}{3} = \\frac{1}{3} \\cdot \\frac{5!}{3!\\,2!} = \\frac{1}{3} \\cdot 10 = \\frac{10}{3} \\]<\/p>\n<p>The full triangle for 5 rows:<\/p>\n<p>\\[ \\begin{array}{ccccc} \\frac{1}{1} &amp; &amp; &amp; &amp; \\\\ \\frac{2}{1} &amp; \\frac{1}{2} &amp; &amp; &amp; \\\\ \\frac{3}{1} &amp; \\frac{3}{2} &amp; \\frac{1}{3} &amp; &amp; \\\\ \\frac{4}{1} &amp; \\frac{3}{1} &amp; \\frac{4}{3} &amp; \\frac{1}{4} &amp; \\\\ \\frac{5}{1} &amp; \\frac{5}{1} &amp; \\frac{10}{3} &amp; \\frac{5}{4} &amp; \\frac{1}{5} \\end{array} \\]<\/p>\n<p><strong>Result:<\/strong> the entry \\(a_{5,3}\\) equals \\(\\frac{10}{3}\\), and the calculator displays all five rows as reduced fractions.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Type the desired number of rows into the <strong>Number of rows<\/strong> field. Any whole number from 1 to 150 is accepted; the default is 5.<\/li>\n<li>The calculator builds the triangle row by row, starting from row 1, and displays every entry as a fraction in lowest terms (for example, 10\/3).<\/li>\n<li>Read the result: row \\(n\\) contains exactly \\(n\\) entries. The first entry of each row equals \\(n\\), and the last entry equals \\(1\/n\\), so you can quickly orient yourself inside the output.<\/li>\n<li>To compare a specific entry with a textbook value, count the rows from the top and the positions from the left \u2014 both start at 1.<\/li>\n<\/ol>\n<h2>What Is the Leibniz Harmonic Triangle?<\/h2>\n<p>The Leibniz Harmonic Triangle is a triangular array of rational numbers named after Gottfried Wilhelm Leibniz, who studied arrays of this kind in the 1670s in connection with the harmonic series. Like <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/floyds-triangle-generator\/\">Floyd's triangle<\/a> or Pascal's triangle, row \\(n\\) contains exactly \\(n\\) entries \u2014 but here every entry is a fraction built from a binomial coefficient.<\/p>\n<p>The link to Pascal's triangle is direct: take the familiar binomial coefficient \\(\\binom{n}{k}\\) and divide it by the position \\(k\\) of the entry within its row. The result is a copy of Pascal's triangle in which each column has been weighted by \\(1\/k\\). The first column therefore contains the natural numbers \\(1, 2, 3, \\dots\\), while the last column contains \\(\\frac{1}{1}, \\frac{1}{2}, \\frac{1}{3}, \\dots\\) \u2014 precisely the terms of the harmonic series, which is where the \"harmonic\" in the name comes from.<\/p>\n<p>Arrays of this type appear in several areas of mathematics: partial-fraction decompositions, telescoping series, generating functions, and integral representations. In the classical reciprocal convention (see the notes below), each entry can even be written as a Beta integral \\(\\int_0^1 x^{k-1}(1-x)^{n-k}\\,dx\\), which connects the triangle to the gamma function and to probability distributions on the unit interval. If your work involves harmonic quantities more generally \u2014 averaging speeds, rates, or resistances \u2014 the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/harmonic-mean\/\">Harmonic Mean calculator<\/a> covers the related statistic.<\/p>\n<p>When you read the calculator's output, note that every fraction is shown in lowest terms: the numerator and denominator are divided by their greatest common divisor. For instance, \\(a_{5,2} = \\frac{1}{2}\\binom{5}{2} = \\frac{10}{2}\\) is displayed as 5\/1, not 10\/2.<\/p>\n<h2>Assumptions, Limitations and Practical Tips<\/h2>\n<ul>\n<li><strong>Indexing:<\/strong> rows and positions are counted from 1, so the entry \\(a_{n,k}\\) exists only for \\(1 \\le k \\le n\\). There is no zeroth column, unlike in some textbook layouts that index from \\(k = 0\\).<\/li>\n<li><strong>Two conventions exist.<\/strong> Many references define the Leibniz harmonic triangle with the reciprocal formula \\(a_{n,k} = \\frac{1}{k\\binom{n}{k}}\\), which produces rows such as \\(\\frac{1}{3}, \\frac{1}{6}, \\frac{1}{3}\\). This calculator uses the convention \\(a_{n,k} = \\frac{1}{k}\\binom{n}{k}\\) shown in the formula above, so check which definition your textbook or problem uses before comparing values.<\/li>\n<li><strong>Whole rows only.<\/strong> The tool generates complete rows, not individual entries. To extract one value, generate enough rows and count to position \\(k\\) in row \\(n\\).<\/li>\n<li><strong>Precision at large rows.<\/strong> Binomial coefficients grow extremely fast: beyond roughly 55\u201360 rows, the central entries exceed what floating-point arithmetic can represent exactly, so very large fractions may lose accuracy. For exact work, stay within about 50 rows.<\/li>\n<li><strong>Display width.<\/strong> With 150 rows the output becomes very wide. If you only need the first few entries of each row, a smaller row count is easier to read.<\/li>\n<\/ul>\n<p>For other classic number arrays and pattern generators, try the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/magic-square-generator\/\">Magic Square generator<\/a> or the Floyd's triangle tool linked above.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Generate the Leibniz Harmonic Triangle for up to 150 rows, with every entry shown as an exact fraction in lowest terms. Just enter the number of rows and read off the values, each computed as (1\/k) \u00d7 C(n,k).<\/p>\n","protected":false},"author":70,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[72],"class_list":["post-61927","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-arithmetic-calculators"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61927","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/70"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=61927"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61927\/revisions"}],"predecessor-version":[{"id":66255,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61927\/revisions\/66255"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=61927"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=61927"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=61927"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}