{"id":343395,"date":"2022-12-31T03:00:37","date_gmt":"2022-12-31T03:00:37","guid":{"rendered":"http:\/\/savepearlharbor.com\/?p=343395"},"modified":"-0001-11-30T00:00:00","modified_gmt":"-0001-11-29T21:00:00","slug":"","status":"publish","type":"post","link":"https:\/\/savepearlharbor.com\/?p=343395","title":{"rendered":"<span>\u041e\u0442 n! \u0434\u043e n^n \u0432\u0441\u0435\u0433\u043e n \u0441\u043b\u0430\u0433\u0430\u0435\u043c\u044b\u0445, \u043d\u043e \u044d\u0442\u043e \u2014 \u043d\u0435 \u0441\u0430\u043c\u043e\u0435 \u0437\u0430\u043c\u0435\u0447\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0435<\/span>"},"content":{"rendered":"<div><\/div>\n<div id=\"post-content-body\">\n<div>\n<div class=\"article-formatted-body article-formatted-body article-formatted-body_version-2\">\n<div xmlns=\"http:\/\/www.w3.org\/1999\/xhtml\">\n<p>\u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0444\u043e\u0440\u043c\u0443\u043b\u0443:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"formula\" source=\"n!=n^n-C_n^1(n-1)^n+C_n^2(n-2)^n-C_n^3(n-3)^n+...+(-1)^{(n-1)}C{_n}^{(n-1)}   (*)\" alt=\"n!=n^n-C_n^1(n-1)^n+C_n^2(n-2)^n-C_n^3(n-3)^n+...+(-1)^{(n-1)}C{_n}^{(n-1)}   (*)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/f41\/415\/7a7\/f414157a7adf17c3c9b932574b2d2384.svg\" width=\"638\" height=\"26\"\/><\/p>\n<p>\u0415\u0441\u043b\u0438 \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433\u0430 \u0432 \u0432\u0438\u0434\u0435 \u0440\u044f\u0434\u0430:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"formula\" source=\"n! = \\sqrt{2\\pi n}(\\frac{n}{e})^n(1+\\frac{1}{12 n}+\\frac{1}{288n^2}-\\frac{139}{51840 n^3}+...)\" alt=\"n! = \\sqrt{2\\pi n}(\\frac{n}{e})^n(1+\\frac{1}{12 n}+\\frac{1}{288n^2}-\\frac{139}{51840 n^3}+...)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/a94\/5cf\/594\/a945cf5941e58bc3e34e08c0399bc2af.svg\" width=\"440\" height=\"43\"\/><\/p>\n<p>\u0412\u0432\u0435\u0434\u0435\u043c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0435 \u043e\u0431\u043e\u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"formula\" source=\"-\\delta(n)=n!-n^n\" alt=\"-\\delta(n)=n!-n^n\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/d08\/516\/32c\/d0851632c7bca6360081b1b30fdda4c9.svg\" width=\"138\" height=\"22\"\/><\/p>\n<p>\u041f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u0444\u043e\u0440\u043c\u0443\u043b\u0443:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"formula\" source=\"n! \\approx \\delta(n)[\\frac{\\sqrt{2\\pi n}(1+\\frac{1}{12 n}+\\frac{1}{288n^2}-\\frac{139}{51840 n^3}+...)}{e^n-\\sqrt{2\\pi n}(1+\\frac{1}{12 n}+\\frac{1}{288n^2}-\\frac{139}{51840 n^3}+...)}] (**)\" alt=\"n! \\approx \\delta(n)[\\frac{\\sqrt{2\\pi n}(1+\\frac{1}{12 n}+\\frac{1}{288n^2}-\\frac{139}{51840 n^3}+...)}{e^n-\\sqrt{2\\pi n}(1+\\frac{1}{12 n}+\\frac{1}{288n^2}-\\frac{139}{51840 n^3}+...)}] (**)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/475\/ea0\/b14\/475ea0b147bd856f4bbec10c1e613ef8.svg\" width=\"483\" height=\"64\"\/><\/p>\n<p>\u041f\u043e\u0441\u0447\u0438\u0442\u0430\u0435\u043c \u0444\u043e\u0440\u043c\u0443\u043b\u0443 (**), \u0434\u043e\u043f\u0443\u0441\u0442\u0438\u043c \u0434\u043b\u044f n=20.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"formula\" source=\"20!-20^{20}=-104857597567097991823360000\" alt=\"20!-20^{20}=-104857597567097991823360000\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/cda\/c3e\/c25\/cdac3ec25a1894e13202cbc4d8dbde77.svg\" width=\"384\" height=\"22\"\/><img loading=\"lazy\" decoding=\"async\" class=\"formula\" source=\"20! \\approx 2.4329028230918*10^{18}\" alt=\"20! \\approx 2.4329028230918*10^{18}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/fd4\/5a6\/867\/fd45a6867a9ebd21e94c01908f02888d.svg\" width=\"244\" height=\"21\"\/><\/p>\n<p>\u041f\u0440\u0438 \u044d\u0442\u043e\u043c \u0442\u043e\u0447\u043d\u043e\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435: 20!=2432902008176640000<\/p>\n<p>\u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u0432\u0438\u0434\u0430 (**) \u043f\u043e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442 \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 \u0440\u044f\u0434\u043e\u043c \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433\u0430 \u0438 \u0440\u044f\u0434\u043e\u043c \u0438\u0437 n \u0441\u043b\u0430\u0433\u0430\u0435\u043c\u044b\u0445 \u0438\u0437 \u043f\u0440\u0430\u0432\u043e\u0439 \u0447\u0430\u0441\u0442\u0438 \u0444\u043e\u0440\u043c\u0443\u043b\u044b (*).<\/p>\n<p>\u0415\u0441\u043b\u0438 \u0436\u0435 \u0443\u0447\u0435\u0441\u0442\u044c, \u0447\u0442\u043e <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"formula\" source=\"n!=(\\frac{n}{e})^n \\sqrt{2 \\pi n}\\sum_{k=0}^{\\infty}\\frac{a_k}{n^k}\" alt=\"n!=(\\frac{n}{e})^n \\sqrt{2 \\pi n}\\sum_{k=0}^{\\infty}\\frac{a_k}{n^k}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/cee\/813\/809\/cee813809e5ecab6ff7e37f0cd74a823.svg\" width=\"196\" height=\"57\"\/><img loading=\"lazy\" decoding=\"async\" class=\"formula\" source=\"a_k=\\sum_{j=1}^{2k}(-1)^j\\frac{d_3(2k+2j,j)}{2^{j+k}(j+k)!},k \\in N, (***)\" alt=\"a_k=\\sum_{j=1}^{2k}(-1)^j\\frac{d_3(2k+2j,j)}{2^{j+k}(j+k)!},k \\in N, (***)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/9c7\/a80\/2b3\/9c7a802b31709f68ed442d8e728b12f6.svg\" width=\"359\" height=\"62\"\/><\/p>\n<p>\u0433\u0434\u0435 <img loading=\"lazy\" decoding=\"async\" class=\"formula inline\" source=\"d_3(n,m)\" alt=\"d_3(n,m)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/1d5\/8b4\/e98\/1d58b4e981c1977ad816485ac3f97147.svg\" width=\"71\" height=\"22\"\/>&#8212; \u043a\u043e\u043b\u0438\u0447\u0435\u0441\u0442\u0432\u043e \u043f\u0435\u0440\u0435\u0441\u0442\u0430\u043d\u043e\u0432\u043e\u043a \u0438\u0437 n \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043e\u0432 \u0441 m \u0446\u0438\u043a\u043b\u0430\u043c\u0438, \u043a\u0430\u0436\u0434\u044b\u0439 \u0438\u0437 \u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0438\u043c\u0435\u0435\u0442 \u0434\u043b\u0438\u043d\u0443 \u043d\u0435 \u043c\u0435\u043d\u0435\u0435 3.<\/p>\n<blockquote>\n<p>Comtet L. Advanced Combinatorics: The Art of Finite and Infinite Expansions. &#8212; D. Reidel Publishing Company, 1974. &#8212; 267 p.<\/p>\n<\/blockquote>\n<p>\u041f\u043e\u043b\u0443\u0447\u0430\u0435\u0442\u0441\u044f \u0437\u0430\u043c\u0435\u0447\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0435 \u0441\u043e\u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 \u043a\u043e\u043d\u0435\u0447\u043d\u043e\u0433\u043e \u0447\u0438\u0441\u043b\u0430 \u0441\u043b\u0430\u0433\u0430\u0435\u043c\u044b\u0445, \u0441\u043e\u0434\u0435\u0440\u0436\u0430\u0449\u0438\u0445 \u0431\u0438\u043d\u043e\u043c\u0438\u0430\u043b\u044c\u043d\u044b\u0435 \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442\u044b \u0438 \u0440\u044f\u0434\u043e\u043c (***).<\/p>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p> <!----> <!----><\/div>\n<p> <!----> <!----><br \/> \u0441\u0441\u044b\u043b\u043a\u0430 \u043d\u0430 \u043e\u0440\u0438\u0433\u0438\u043d\u0430\u043b \u0441\u0442\u0430\u0442\u044c\u0438 <a href=\"https:\/\/habr.com\/ru\/post\/708742\/\"> https:\/\/habr.com\/ru\/post\/708742\/<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<div><\/div>\n<div id=\"post-content-body\">\n<div>\n<div class=\"article-formatted-body article-formatted-body article-formatted-body_version-2\">\n<div xmlns=\"http:\/\/www.w3.org\/1999\/xhtml\">\n<p>\u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0444\u043e\u0440\u043c\u0443\u043b\u0443:<\/p>\n<p>\u0415\u0441\u043b\u0438 \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433\u0430 \u0432 \u0432\u0438\u0434\u0435 \u0440\u044f\u0434\u0430:<\/p>\n<p>\u0412\u0432\u0435\u0434\u0435\u043c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0435 \u043e\u0431\u043e\u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f:<\/p>\n<p>\u041f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u0444\u043e\u0440\u043c\u0443\u043b\u0443:<\/p>\n<p>\u041f\u043e\u0441\u0447\u0438\u0442\u0430\u0435\u043c \u0444\u043e\u0440\u043c\u0443\u043b\u0443 (**), \u0434\u043e\u043f\u0443\u0441\u0442\u0438\u043c \u0434\u043b\u044f n=20.<\/p>\n<p>\u041f\u0440\u0438 \u044d\u0442\u043e\u043c \u0442\u043e\u0447\u043d\u043e\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435: 20!=2432902008176640000<\/p>\n<p>\u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u0432\u0438\u0434\u0430 (**) \u043f\u043e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442 \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 \u0440\u044f\u0434\u043e\u043c \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433\u0430 \u0438 \u0440\u044f\u0434\u043e\u043c \u0438\u0437 n \u0441\u043b\u0430\u0433\u0430\u0435\u043c\u044b\u0445 \u0438\u0437 \u043f\u0440\u0430\u0432\u043e\u0439 \u0447\u0430\u0441\u0442\u0438 \u0444\u043e\u0440\u043c\u0443\u043b\u044b (*).<\/p>\n<p>\u0415\u0441\u043b\u0438 \u0436\u0435 \u0443\u0447\u0435\u0441\u0442\u044c, \u0447\u0442\u043e <\/p>\n<p>\u0433\u0434\u0435 &#8212; \u043a\u043e\u043b\u0438\u0447\u0435\u0441\u0442\u0432\u043e \u043f\u0435\u0440\u0435\u0441\u0442\u0430\u043d\u043e\u0432\u043e\u043a \u0438\u0437 n \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043e\u0432 \u0441 m \u0446\u0438\u043a\u043b\u0430\u043c\u0438, \u043a\u0430\u0436\u0434\u044b\u0439 \u0438\u0437 \u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0438\u043c\u0435\u0435\u0442 \u0434\u043b\u0438\u043d\u0443 \u043d\u0435 \u043c\u0435\u043d\u0435\u0435 3.<\/p>\n<blockquote>\n<p>Comtet L. Advanced Combinatorics: The Art of Finite and Infinite Expansions. &#8212; D. Reidel Publishing Company, 1974. &#8212; 267 p.<\/p>\n<\/blockquote>\n<p>\u041f\u043e\u043b\u0443\u0447\u0430\u0435\u0442\u0441\u044f \u0437\u0430\u043c\u0435\u0447\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0435 \u0441\u043e\u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 \u043a\u043e\u043d\u0435\u0447\u043d\u043e\u0433\u043e \u0447\u0438\u0441\u043b\u0430 \u0441\u043b\u0430\u0433\u0430\u0435\u043c\u044b\u0445, \u0441\u043e\u0434\u0435\u0440\u0436\u0430\u0449\u0438\u0445 \u0431\u0438\u043d\u043e\u043c\u0438\u0430\u043b\u044c\u043d\u044b\u0435 \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442\u044b \u0438 \u0440\u044f\u0434\u043e\u043c (***).<\/p>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p> <!----> <!----><\/div>\n<p> <!----> <!----><br \/> \u0441\u0441\u044b\u043b\u043a\u0430 \u043d\u0430 \u043e\u0440\u0438\u0433\u0438\u043d\u0430\u043b \u0441\u0442\u0430\u0442\u044c\u0438 <a href=\"https:\/\/habr.com\/ru\/post\/708742\/\"> https:\/\/habr.com\/ru\/post\/708742\/<\/a><br \/><\/br><\/br><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"class_list":["post-343395","post","type-post","status-publish","format-standard","hentry"],"_links":{"self":[{"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=\/wp\/v2\/posts\/343395","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=343395"}],"version-history":[{"count":0,"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=\/wp\/v2\/posts\/343395\/revisions"}],"wp:attachment":[{"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=343395"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=343395"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=343395"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}