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(a,b)-Pascal triangle
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The (a,b)-Pascal triangle is a geometric arrangement of numbers produced recursively in the same way as for the original Pascal triangle, except for its boundary conditions (for the leftmost and rightmost entries on each rows.) Assuming its boundary conditions to be positive integers, the rightmost nonzero entries are initialized to b, b > 0, and its leftmost nonzero entries (except the first row for n = 0) are initialized to a, a > 0. The (a,b)-Pascal triangle can be built by superimposing the (k,1)-Pascal triangle with k = a-1 and the (1,k)-Pascal triangle with k = b-1, i.e.:
- .
The rectangular version of the (a,b)-Pascal triangle n = 0 b 1 a b 2 a b 3 a b 4 a b 5 a b 6 a b 7 a b 8 a b 9 a b 10 a b 11 a b 12 a b j = 0 1 2 3 4 5 6 7 8 9 10 11 12
In the equilateral version of the (a,b)-Pascal triangle, we start with a cell (row 0) initialized to b, with the leftmost nonzero cell in the row below initialized to a, in a staggered array of empty (0) cells. We then recursively evaluate the cells as the sum of the two cells staggered above. The triangle thus grows into an equilateral triangle.
In the rectangular version of the (a,b)-Pascal triangle, we start with a cell (row 0) initialized to b, with the cell below it initialized to a, in a regular array of empty (0) cells. We then recursively evaluate the cells as the sum of the one above left and the one directly above. The triangle thus grows into a rectangular triangle.
If a and b are positive numbers, then all the interior cells are necessarily greater than or equal to a + b. The number of cells from rows 0 to n which are equal to a is thus n. (Cf. A000027(n),) the number of cells from rows 0 to n which are equal to b is thus n+1 (Cf. A000027(n+1),) and the number of cells from rows 0 to n which are greater than or equal to a + b is , the (n-1)^{th} triangular number.
For comparison, the (1,1)-Pascal triangle (Pascal's triangle) is shown below:
The rectangular version of Pascal's triangle
(Figurate Number Triangle)^{[1]}n = 0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1 6 1 6 15 20 15 6 1 7 1 7 21 35 35 21 7 1 8 1 8 28 56 70 56 28 8 1 9 1 9 36 84 126 126 84 36 9 1 10 1 10 45 120 210 252 210 120 45 10 1 11 1 11 55 165 330 462 462 330 165 55 11 1 12 1 12 66 220 495 792 924 792 495 220 66 12 1 j = 0 1 2 3 4 5 6 7 8 9 10 11 12
Recursion rule
The (a,b)-Pascal triangle recursion rule is:
Formulae
where when n < 0, r < 0 or n - r < 0,^{[2]} and is cell (n, j) of Pascal's triangle.
(a,b)-Pascal triangle formulae and additive commutative semigroups
The rectangular version of the (a,b)-Pascal triangle n = 0 b 1 a b 2 a b 3 a b 4 a b 5 a b 6 a b 7 a b 8 a b 9 a b 10 a b 11 a b 12 a b j = 0 1 2 3 4 5 6 7 8 9 10 11 12
Since the binomial coefficients are counting devices (telling how many times that something was added,) we only need a and b to belong to an additive commutative semigroup ^{[3]} (a semigroup being a set with an operation having the associativity and closure properties.)
Both associativity and commutativity are required to allow for the grouping of the a's and b's:
Therefore a and b could both belong to, among others:
- Natural numbers
- Rational integers
- Rational integers (mod n)
- Rational numbers
- Algebraic integers (e.g. Gaussian integers )
- Algebraic numbers
- Real numbers
- Complex numbers (equivalent to two (a,b)-Pascal triangles: one for the real parts, one for the imaginary parts)
- Quaternions (equivalent to four (a,b)-Pascal triangles)
- Octonions (equivalent to eight (a,b)-Pascal triangles)
- Sedenions (equivalent to sixteen (a,b)-Pascal triangles)
- Vectors (tensors of rank 1) in or over any given field (equivalent to n (a,b)-Pascal triangles)
- Matrices (tensors of rank 2) in or over any given field (equivalent to m × n (a,b)-Pascal triangles)
- Tensors with same rank r in or over any given field (equivalent to (a,b)-Pascal triangles)
- ...
(a,b)-Pascal triangle rows
(a,b)-Pascal triangle rows sums
(a,b)-Pascal triangle rows alternating sign sums
(a,b)-Pascal (rectangular) triangle columns
Table of columns sequences
Table of columns sequences related formulae
(a,b)-Pascal (rectangular) triangle falling diagonals
Table of falling diagonals sequences
Table of falling diagonals sequences related formulae
(a,b)-Pascal (rectangular) triangle rising diagonals
(a,b)-Pascal (rectangular) triangle rising diagonals sums
(a,b)-Pascal (rectangular) triangle rising diagonals alternating sign sums
(a,b)-Pascal triangle central elements
The central elements (for row 2m, m ≥ 0) of the (a,b)-Pascal triangle give the sequence:
- {b, a+b, 3a+3b, 10a+10b, 35a+35b, 126a+126b, 462a+462b, ...}
which is given by the formulae:
where:
is the m^{th} Catalan number (also called Segner numbers) (Cf. A000108(m).)
The generating function is:
where is the generating function of the Catalan numbers:
Generalizations of the (a,b)-Pascal triangle
Keeping the original Pascal triangle recurrence rule unchanged for the interior cells of the triangle, then one might want to consider a generalized (a(n),b(n))-Pascal triangle, n ≥ 0, where a_{n} ≡ a(n) and b_{n} ≡ b(n) are integer sequences.
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
Notes
- ↑ Weisstein, Eric W., Figurate Number Triangle, From MathWorld--A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Binomial Coefficient, From MathWorld--A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Semigroup, From MathWorld--A Wolfram Web Resource.
External links
- S. Plouffe, Approximations de Séries Génératrices et Quelques Conjectures, Dissertation, Université du Québec à Montréal, 1992.
- S. Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
- Herbert S. Wilf, generatingfunctionology, 2^{nd} ed., 1994.