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==Треугольник чисел Эйлера I рода и явная формула==
===Явная формула===
Приведем также без вывода явную формулу для вычисления чисел Эйлера I рода:
<tex dpi = "180">\left\langle{n\atop m}\right\rangle = \sum_{k=0}^{m}[ (-1)^k {n+1\choose k} (m+1-k)^n]</tex>
===Треугольник чисел Эйлера I рода===
На значениях <tex dpi = "130">n = m</tex> чисел Эйлера I рода можно построить массив <tex dpi = "130">n \times m</tex>, нижнедиагональная часть которого названа треугольником чисел Эйлера I рода.
::{| class="number_triangle"
|- align="center"
| style="background:white; color:black; width:50px;" |
| style="background:white; color:black; width:50px;" | '''''k = 0'''''
| style="background:white; color:black; width:50px;" | '''''1'''''
| style="background:white; color:black; width:50px;" | '''''2'''''
| style="background:white; color:black; width:50px;" | '''''3'''''
| style="background:white; color:black; width:50px;" | '''''4'''''
| style="background:white; color:black; width:50px;" | '''''5'''''
| style="background:white; color:black; width:50px;" | '''''6'''''
| style="background:white; color:black; width:50px;" | '''''7'''''
| style="background:white; color:black; width:50px;" | '''''8'''''
| style="background:white; color:black; width:50px;" | '''''9'''''
| style="background:white; color:black; width:50px;" | '''''10'''''
| style="background:white; color:black; width:50px;" | '''''11'''''
| style="background:white; color:black; width:50px;" | '''''12'''''
|- align="center"
| style="background:white; color:black;" | '''''n = 0'''''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
|- align="center"
| style="background:white; color:black;" | '''''1'''''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
|- align="center"
| style="background:white; color:black;" | '''''2'''''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
|- align="center"
| style="background:white; color:black;" | '''''3'''''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:black;" | '''4'''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
|- align="center"
| style="background:white; color:black;" | '''''4'''''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:black;" | '''11'''
| style="background:#FFDEAD; color:black;" | '''11'''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
|- align="center"
| style="background:white; color:black;" | '''''5'''''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:black;" | '''26'''
| style="background:#FFDEAD; color:black;" | '''66'''
| style="background:#FFDEAD; color:black;" | '''26'''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
|- align="center"
| style="background:white; color:black;" | '''''6'''''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:black;" | '''57'''
| style="background:#FFDEAD; color:black;" | '''302'''
| style="background:#FFDEAD; color:black;" | '''302'''
| style="background:#FFDEAD; color:black;" | '''57'''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
|- align="center"
| style="background:white; color:black;" | '''''7'''''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:black;" | '''120'''
| style="background:#FFDEAD; color:black;" | '''1191'''
| style="background:#FFDEAD; color:black;" | '''2416'''
| style="background:#FFDEAD; color:black;" | '''1191'''
| style="background:#FFDEAD; color:black;" | '''120'''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
|- align="center"
| style="background:white; color:black;" | '''''8'''''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:black;" | '''247'''
| style="background:#FFDEAD; color:black;" | '''4293'''
| style="background:#FFDEAD; color:black;" | '''15619'''
| style="background:#FFDEAD; color:black;" | '''15619'''
| style="background:#FFDEAD; color:black;" | '''4293'''
| style="background:#FFDEAD; color:black;" | '''247'''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
|- align="center"
| style="background:white; color:black;" | '''''9'''''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:black;" | '''502'''
| style="background:#FFDEAD; color:black;" | '''14608'''
| style="background:#FFDEAD; color:black;" | '''88234'''
| style="background:#FFDEAD; color:black;" | '''156190'''
| style="background:#FFDEAD; color:black;" | '''88234'''
| style="background:#FFDEAD; color:black;" | '''14608'''
| style="background:#FFDEAD; color:black;" | '''502'''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
|- align="center"
| style="background:white; color:black;" | '''''10'''''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:black;" | '''1013'''
| style="background:#FFDEAD; color:black;" | '''47840'''
| style="background:#FFDEAD; color:black;" | '''455192'''
| style="background:#FFDEAD; color:black;" | '''1310354'''
| style="background:#FFDEAD; color:black;" | '''1310354'''
| style="background:#FFDEAD; color:black;" | '''455192'''
| style="background:#FFDEAD; color:black;" | '''47840'''
| style="background:#FFDEAD; color:black;" | '''1013'''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
|- align="center"
| style="background:white; color:black;" | '''''11'''''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:black;" | '''2036'''
| style="background:#FFDEAD; color:black;" | '''152637'''
| style="background:#FFDEAD; color:black;" | '''2203488'''
| style="background:#FFDEAD; color:black;" | '''9738114'''
| style="background:#FFDEAD; color:black;" | '''15724248'''
| style="background:#FFDEAD; color:black;" | '''9738114'''
| style="background:#FFDEAD; color:black;" | '''2203488'''
| style="background:#FFDEAD; color:black;" | '''152637'''
| style="background:#FFDEAD; color:black;" | '''2036'''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:red;" | '''0'''
| style="background:#FFDEAD; color:red;" | '''0'''
|- align="center"
| style="background:white; color:black;" | '''''12'''''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:black;" | '''4083'''
| style="background:#FFDEAD; color:black;" | '''478271'''
| style="background:#FFDEAD; color:black;" | '''10187685'''
| style="background:#FFDEAD; color:black;" | '''66318474'''
| style="background:#FFDEAD; color:black;" | '''162512286'''
| style="background:#FFDEAD; color:black;" | '''162512286'''
| style="background:#FFDEAD; color:black;" | '''66318474'''
| style="background:#FFDEAD; color:black;" | '''10187685'''
| style="background:#FFDEAD; color:black;" | '''478271'''
| style="background:#FFDEAD; color:black;" | '''4083'''
| style="background:#FFDEAD; color:black;" | '''1'''
| style="background:#FFDEAD; color:red;" | '''0'''
|}
Стоит отметить, что гистрограмма, построенная на значениях чисел Эйлера I рода аппроксимируется к гистограмме, построенной на биноминальных коээфициентах (оба графика, представленные '''справа''', смасштабированы; масштаб указан на гистограмме):
[[Файл:Euler_I_hist.gif|300px|thumb|right|Числа Эйлера I рода (m < 90)]]
[[Файл:Binomial_hist.gif|300px|thumb|right|Биномиальные коээфициенты (m < 60)]]
'''Полезные факты о числах Эйлера I рода'''
1. Нетрудно увидеть, что каждый ряд ненулевых значений симметричен относительно своей середины, то есть:
:<tex dpi = "160">\left\langle{n\atop m}\right\rangle = \left\langle{n\atop (n-1) - k}\right\rangle,\ n \ge 1,\ 0 \le k \le n-1. \, </tex>
2. Сумма всех значений каждого ряда равна <tex dpi = "130"> n! </tex>:
:<tex dpi = "160">\sum_{k=0}^{n} \left\langle{n\atop m}\right\rangle = n!,\ n \ge 0, \,</tex>
3. <tex dpi = "160">\sum_{m=0}^n (-1)^m {\left\langle{n\atop m}\right\rangle}{n-1\choose m}^{-1}=0.</tex>
==Числа Эйлера II рода==
