User:Tawfiq-school

Factorial
$$n!=n\times{(n-1)}\times{\cdots}\times{2}\times{1}$$

$$2^n\prod_{k=1}^{n}T_{2n}=(2n+1)!$$

Binomial coefficient
$$\sum_{r=0}^{k}{k\choose r}^2={2k\choose k}$$

$$\sum_{r=0}^{2k}(-1)^r{2k\choose r}^2=(-1)^k{2k\choose k}$$

Factorial
$$3!!-2!=1^2\cdots(1)$$

$$5!!-3!=3^2\cdots(2)$$

$$7!!-4!=9^2\cdots(3)$$

$$9!!-6!=15^2\cdots(4)$$

Special numbers
Triangular number

Tn = 1, 3 , 6 , 10 , 15 , 21 , ...

Tetrahedral number

Te(n) = 1, 4 , 10 , 20 , 35 , 56 , ...

Square pyramidal number

Pn = 1, 5 , 14 , 30 , 55 , 91 , ...

Octagonal number

On = 1, 8 , 21 , 40 , 65 , ...

Below are their relatioship

$$nP_n-(P_{n-1}+P_{n-2}+...+P_1)=T_n^2$$

Short hand notation

$$nP_n-\sum_{k=1}^{n-1}P_k=T_n^2$$

E.g

$$6\times{91}-(55+30+14+5+1)=21^2$$

$$Te_{n}+Te_{n-1}=P_{n}\cdots(1)$$

$$(n+1)^3-P_{n+1}-(n^3-P_n)=T_{2n}\cdots(2)$$

$$Te_n-Te_{n-2}=n^2\cdots(3)$$

$$P_n-P_{n-1}=T_n+T_{n-1}\cdots(4)$$

$$P_n+T_n=2Te_n\cdots(5)$$

$$(n+1)^3+T_{n}^2=T_{n+1}^2$$

Binomial coefficients

$$1\cdot[(n+1)^2-T_{n+1}]-1\cdot(n^2-T_n)=n\cdots(6)$$

$$1\cdot[(n+2)^3-T_{n+2}]-2\cdot[(n+1)^3-T_{n+1}]+1\cdot(n^3-T_n)=6n+5\cdots(7)$$

$$1\cdot[(n+2)^3-Te_{n+2}]-2\cdot[(n+1)^3-Te_{n+1}]+1\cdot(n^3-Te_n)=5n+4\cdots(8)$$

$$1\cdot[(n+2)^3-P_{n+2}]-2\cdot[(n+1)^3-P_{n+1}]+1\cdot(n^3-P_n)=4n+3\cdots(9)$$

Octagonal and Triangular numbers

$$O_{n+2}-2O_{n+1}+O_n-T_{n+2}+2T_{n+1}-T_n=5\cdots(10)$$

$$O_{n+3}-3O_{n+2}+3O_{n+1}-O_n=0\cdots(11)$$

$$O_{n+4}-4O_{n+3}+6O_{n+2}-4O_{n+1}+O_n=0\cdots(12)$$

$${4\choose 0}O_{n+4}-{4\choose 1}O_{n+3}+{4\choose 2}O_{n+2}-{4\choose 3}O_{n+1}+ {4\choose 4}O_n=0$$

General

Sn = {T(n), Te(n) , O(n) , P(n)} 

$$\sum_{r=0}^{k}(-1)^r{k\choose r}S_{n+k-r}=0$$

$$\sum_{r=0}^{k}(-1)^r{k\choose r}T_{n+k-r}=0$$

$$\sum_{r=0}^{k}(-1)^r{k\choose r}Te_{n+k-r}=0$$

etc.

Where k ≥ 3

Sum of Triangular number

$$\sum_{r=1}^{2n}(-1)^rT_r^2=8T_n^2$$

Product

$$\frac{T_{2n}T_{2n+1}}{T_{2n}+T_{2n+1}}=2T_{n}$$

$$\frac{T_{n+1}}{T_{n}}=\frac{n+2}{n}$$

$$nT_{n+1}-(n+1)T_{n}=\frac{n(n+1)}{2}$$

$$nT_{n+2k}-(n+2k)T_{n}=kn(n+2k)\cdots(1)$$