%0 Journal Article
%T Expansion of the Shape of Numbers
%A Ji Peng
%J Open Access Library Journal
%V 8
%N 1
%P 1-18
%@ 2333-9721
%D 2021
%I Open Access Library
%R 10.4236/oalib.1107120
%X This article extends the concept of the shape of numbers. Originally, a shape was defined as [1, <i>K</i><sub>1</sub>, <i>K</i><sub>2</sub>, &middot;&middot;&middot;], 1<<i>K</i><sub>1</sub><<i>K</i><sub>2</sub>< &middot;&middot;&middot;, <i>K</i><sub>i</sub>&isin;<i>N</i>. In this paper, the domain of a shape is extended from <i>N</i> to <i>Z</i>, the low bound is extended from 1 to <i>Z</i>, and <i>K</i><sub>i</sub><<i>K</i><sub>i 1</sub>, <i>K</i><sub>i</sub>=<i>K</i><sub>i 1</sub>, <i>K</i><sub>i</sub>><i>K</i><sub>i 1</sub> are allowed, which prove that they can be calculated with the similar form (<i>T</i><sub>0</sub> <i>K</i><sub>0</sub>)(<i>T</i><sub>1</sub> <i>K</i><sub>1</sub>)(<i>T</i><sub>2</sub> <i>K</i><sub>2</sub>) &middot;&middot;&middot;. In this way, a lot of calculation formulas can be obtained. At the end, the form is obtained to calculate <i>K</i><sub>1</sub>x&middot;&middot;&middot;x<i>K</i><sub>M</sub> (<i>L</i> <i>K</i><sub>1</sub>)x&middot;&middot;&middot;x(<i>L</i> <i>K</i><sub>M</sub>) (<i>2L</i> <i>K</i><sub>1</sub>)x&middot;&middot;&middot;x(<i>2L</i> <i>K</i><sub>M</sub>) (<i>3L</i> <i>K</i><sub>1</sub>)x&middot;&middot;&middot;x(<i>3L</i> <i>K</i><sub>M</sub>) &middot;&middot;&middot;.
%K Shape of Numbers
%K Calculation Formula
%K Combinatorics
%K Congruence
%K Stirling Number
%U http://www.oalib.com/paper/6325697