(a(n),b(n))-Pascal triangle

The (a(n),b(n))-Pascal triangle, or (a_n,b_n)-Pascal triangle, is a generalization of the (a,b)-Pascal triangle where a_n ≡ a(n) and b_n ≡ b(n) are integer sequences, keeping the original Pascal triangle recurrence rule unchanged for the interior cells of the triangle.

The rectangular version of the (a_n,b_n)-Pascal triangle

n = 0

b0

1

a1

b1

2

a2

$\scriptstyle \sum _{i=1}^{1}{\binom {1-i}{0}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{1}{\binom {1-i}{0}}b_{i}\,$

b2

3

a3

$\scriptstyle \sum _{i=1}^{2}{\binom {2-i}{0}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{1}{\binom {2-i}{1}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{1}{\binom {2-i}{1}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{2}{\binom {2-i}{0}}b_{i}\,$

b3

4

a4

$\scriptstyle \sum _{i=1}^{3}{\binom {3-i}{0}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{1}{\binom {3-i}{2}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{2}{\binom {3-i}{1}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{2}{\binom {3-i}{1}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{1}{\binom {3-i}{2}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{3}{\binom {3-i}{0}}b_{i}\,$

b4

5

a5

$\scriptstyle \sum _{i=1}^{4}{\binom {4-i}{0}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{1}{\binom {4-i}{3}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{3}{\binom {4-i}{1}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{2}{\binom {4-i}{2}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{2}{\binom {4-i}{2}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{3}{\binom {4-i}{1}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{1}{\binom {4-i}{3}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{4}{\binom {4-i}{0}}b_{i}\,$

b5

6

a6

$\scriptstyle \sum _{i=1}^{5}{\binom {5-i}{0}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{1}{\binom {5-i}{4}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{4}{\binom {5-i}{1}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{2}{\binom {5-i}{3}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{3}{\binom {5-i}{2}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{3}{\binom {5-i}{2}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{2}{\binom {5-i}{3}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{4}{\binom {5-i}{1}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{1}{\binom {5-i}{4}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{5}{\binom {5-i}{0}}b_{i}\,$

b6

7

a7

$\scriptstyle \sum _{i=1}^{6}{\binom {6-i}{0}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{1}{\binom {6-i}{5}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{5}{\binom {6-i}{1}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{2}{\binom {6-i}{4}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{4}{\binom {6-i}{2}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{3}{\binom {6-i}{3}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{3}{\binom {6-i}{3}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{4}{\binom {6-i}{2}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{2}{\binom {6-i}{4}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{5}{\binom {6-i}{1}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{1}{\binom {6-i}{5}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{6}{\binom {6-i}{0}}b_{i}\,$

b7

8

a8

$\scriptstyle \sum _{i=1}^{7}{\binom {7-i}{0}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{1}{\binom {7-i}{6}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{6}{\binom {7-i}{1}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{2}{\binom {7-i}{5}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{5}{\binom {7-i}{2}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{3}{\binom {7-i}{4}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{4}{\binom {7-i}{3}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{4}{\binom {7-i}{3}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{3}{\binom {7-i}{4}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{5}{\binom {7-i}{2}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{2}{\binom {7-i}{5}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{6}{\binom {7-i}{1}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{1}{\binom {7-i}{6}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{7}{\binom {7-i}{0}}b_{i}\,$

b8

9

a9

$\scriptstyle \sum _{i=1}^{8}{\binom {8-i}{0}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{1}{\binom {8-i}{7}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{7}{\binom {8-i}{1}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{2}{\binom {8-i}{6}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{6}{\binom {8-i}{2}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{3}{\binom {8-i}{5}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{5}{\binom {8-i}{3}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{4}{\binom {8-i}{4}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{4}{\binom {8-i}{4}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{5}{\binom {8-i}{3}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{3}{\binom {8-i}{5}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{6}{\binom {8-i}{2}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{2}{\binom {8-i}{6}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{7}{\binom {8-i}{1}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{1}{\binom {8-i}{7}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{8}{\binom {8-i}{0}}b_{i}\,$

b9

10

a10

$\scriptstyle \sum _{i=1}^{9}{\binom {9-i}{0}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{1}{\binom {9-i}{8}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{8}{\binom {9-i}{1}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{2}{\binom {9-i}{7}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{7}{\binom {9-i}{2}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{3}{\binom {9-i}{6}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{6}{\binom {9-i}{3}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{4}{\binom {9-i}{5}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{5}{\binom {9-i}{4}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{5}{\binom {9-i}{4}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{4}{\binom {9-i}{5}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{6}{\binom {9-i}{3}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{3}{\binom {9-i}{6}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{7}{\binom {9-i}{2}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{2}{\binom {9-i}{7}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{8}{\binom {9-i}{1}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{1}{\binom {9-i}{8}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{9}{\binom {9-i}{0}}b_{i}\,$

b10

11

a11

$\scriptstyle \sum _{i=1}^{10}{\binom {10-i}{0}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{1}{\binom {10-i}{9}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{9}{\binom {10-i}{1}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{2}{\binom {10-i}{8}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{8}{\binom {10-i}{2}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{3}{\binom {10-i}{7}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{7}{\binom {10-i}{3}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{4}{\binom {10-i}{6}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{6}{\binom {10-i}{4}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{5}{\binom {10-i}{5}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{5}{\binom {10-i}{5}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{6}{\binom {10-i}{4}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{4}{\binom {10-i}{6}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{7}{\binom {10-i}{3}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{3}{\binom {10-i}{7}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{8}{\binom {10-i}{2}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{2}{\binom {10-i}{8}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{9}{\binom {10-i}{1}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{1}{\binom {10-i}{9}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{10}{\binom {10-i}{0}}b_{i}\,$

b11

12

a12

$\scriptstyle \sum _{i=1}^{11}{\binom {11-i}{0}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{1}{\binom {11-i}{10}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{10}{\binom {11-i}{1}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{2}{\binom {11-i}{9}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{9}{\binom {11-i}{2}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{3}{\binom {11-i}{8}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{8}{\binom {11-i}{3}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{4}{\binom {11-i}{7}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{7}{\binom {11-i}{4}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{5}{\binom {11-i}{6}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{6}{\binom {11-i}{5}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{6}{\binom {11-i}{5}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{5}{\binom {11-i}{6}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{7}{\binom {11-i}{4}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{4}{\binom {11-i}{7}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{8}{\binom {11-i}{3}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{3}{\binom {11-i}{8}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{9}{\binom {11-i}{2}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{2}{\binom {11-i}{9}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{10}{\binom {11-i}{1}}b_{i}\,$

$\scriptstyle \sum _{i=1}^{1}{\binom {11-i}{10}}a_{i}+\,$ $\scriptstyle \sum _{i=1}^{11}{\binom {11-i}{0}}b_{i}\,$

b12

j = 0

1

2

3

4

5

6

7

8

9

10

11

12

Recursion rule

The (a_n,b_n)-Pascal triangle recursion rule is:

$T_{(a_{n},b_{n})}(n,n)=b_{n},\ n\geq 0,\,$

$T_{(a_{n},b_{n})}(n,0)=a_{n},\ n\geq 1,\,$

$T_{(a_{n},b_{n})}(n,j)=T_{(a_{n},b_{n})}(n-1,j-1)+T_{(a_{n},b_{n})}(n-1,j),\ 0<j<n.\,$

Formulae

$T_{(a_{n},b_{n})}(n,n)=b_{n},\ n\geq 0,\,$

$T_{(a_{n},b_{n})}(n,0)=a_{n},\ n\geq 1,\,$

$T_{(a_{n},b_{n})}(n,j)=\sum _{i=1}^{n-j}{\binom {n-1-i}{j-1}}a_{i}+\sum _{i=1}^{j}{\binom {n-1-i}{n-1-j}}b_{i},\ 0<j<n.\,$

Multiplicative (p(2n),p(2n+1))-Pascal triangle giving a Gödel encoding of coefficients

Multiplicative (p(2n),p(2n+1))-Pascal triangle giving a Gödel encoding of coefficients

n = 0

$2\,$

1

$3\,$

$5\,$

2

$7\,$

$\scriptstyle 3\cdot 5\,$

$11\,$

3

$13\,$

$\scriptstyle 3\cdot 5\cdot 7\,$

$\scriptstyle 3\cdot 5\cdot 11\,$

$17\,$

4

$19\,$

$\scriptstyle 3\cdot 5\cdot 7\cdot \,$ $\scriptstyle 13\,$

$\scriptstyle 3^{2}\cdot 5^{2}\cdot 7\cdot \,$ $\scriptstyle 11\,$

$\scriptstyle 3\cdot 5\cdot 11\cdot \,$ $\scriptstyle 17\,$

$23\,$

5

$29\,$

$\scriptstyle 3\cdot 5\cdot 7\cdot \,$ $\scriptstyle 13\cdot 19\,$

$\scriptstyle 3^{3}\cdot 5^{3}\cdot 7^{2}\cdot \,$ $\scriptstyle 11\cdot 13\,$

$\scriptstyle 3^{3}\cdot 5^{3}\cdot 7\cdot \,$ $\scriptstyle 11^{2}\cdot 17\,$

$\scriptstyle 3\cdot 5\cdot 11\cdot \,$ $\scriptstyle 17\cdot 23\,$

$31\,$

6

$37\,$

$\scriptstyle 3\cdot 5\cdot 7\cdot \,$ $\scriptstyle 13\cdot 19\cdot 29\,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$41\,$

7

$43\,$

$\scriptstyle 3\cdot 5\cdot 7\cdot \,$ $\scriptstyle 13\cdot 19\cdot 29\cdot \,$ $\scriptstyle 37\,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$47\,$

8

$53\,$

$\scriptstyle 3\cdot 5\cdot 7\cdot \,$ $\scriptstyle 13\cdot 19\cdot 29\cdot \,$ $\scriptstyle 37\cdot 43\,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$59\,$

9

$61\,$

$\scriptstyle 3\cdot 5\cdot 7\cdot \,$ $\scriptstyle 13\cdot 19\cdot 29\cdot \,$ $\scriptstyle 37\cdot 43\cdot 53\,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$67\,$

10

$71\,$

$\scriptstyle 3\cdot 5\cdot 7\cdot \,$ $\scriptstyle 13\cdot 19\cdot 29\cdot \,$ $\scriptstyle 37\cdot 43\cdot 53\cdot \,$ $\scriptstyle 61\,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$73\,$

11

$79\,$

$\scriptstyle 3\cdot 5\cdot 7\cdot \,$ $\scriptstyle 13\cdot 19\cdot 29\cdot \,$ $\scriptstyle 37\cdot 43\cdot 53\cdot \,$ $\scriptstyle 61\cdot 71\,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$83\,$

12

$89\,$

$\scriptstyle 3\cdot 5\cdot 7\cdot \,$ $\scriptstyle 13\cdot 19\cdot 29\cdot \,$ $\scriptstyle 37\cdot 43\cdot 53\cdot \,$ $\scriptstyle 61\cdot 71\cdot 79\,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$\scriptstyle \,$

$97\,$

j = 0

1

2

3

4

5

6

7

8

9

10

11

12

The multiplicative (p_2n,p_2n+1)-Pascal triangle gives a Gödel encoding of the (a_n,b_n)-Pascal triangle, i.e.:

$M_{(p_{2n},p_{2n+1})}(n,j)=\prod _{i=1}^{2n+1}{p_{i}}^{\alpha _{i}}\,$

where the exponents $\scriptstyle \alpha _{i}\,$ are the coefficients in

$T_{(a_{n},b_{n})}(n,j)=\sum _{i=1}^{2n+1}{\alpha _{i}}{c_{i}},\,$

where

$c_{2k+1}=b_{k},\ k\geq 0,\,$

$c_{2k}=a_{k},\ k\geq 1.\,$

For example:

$M_{(p_{2n},p_{2n+1})}(5,2)=3^{3}\cdot 5^{3}\cdot 7^{2}\cdot 11\cdot 13\,$

corresponds to

$T_{(a_{n},b_{n})}(5,2)=3a_{1}+3b_{1}+2a_{2}+b_{2}+a_{3}\,$

Multiplicative recursion rule

The multiplicative (p_2n,p_2n+1)-Pascal triangle recursion rule is:

$M_{(p_{2n},p_{2n+1})}(n,n)=p_{2n+1},\ n\geq 0,\,$

$M_{(p_{2n},p_{2n+1})}(n,0)=p_{2n},\ n\geq 1,\,$

$M_{(p_{2n},p_{2n+1})}(n,j)=M_{(p_{2n},p_{2n+1})}(n-1,j-1)\cdot M_{(p_{2n},p_{2n+1})}(n-1,j),\ 0<j<n,\,$

where $\scriptstyle p_{i},\ i\geq 1\,$ is the i^th prime.

See also

• (1,1)-Pascal triangle or Pascal's triangle
• (1,2)-Pascal triangle or Lucas triangle
• (1,k)-Pascal triangle
• (2,1)-Pascal triangle
• (k,1)-Pascal triangle
• (a,b)-Pascal triangle
• (a(n),b(n))-Pascal triangle