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(1,k)Pascal triangle
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The (1,k)Pascal triangle has its rightmost nonzero entries initialized to k and its leftmost nonzero entries (except the first row for n = 0) initialized to 1. Thus the rows of the (1,k)Pascal triangle are the leftright reversal of the rows of the (k,1)Pascal triangle, with the exception of the first row (for n = 0) which is now k instead of 1.
The (1,k)Pascal triangle is a geometric arrangement of numbers produced recursively which generates (in its falling interior diagonals starting from the rightmost one) the (k+2)gonal gnomonic numbers, the (k+2)gonal numbers, the (k+2)gonal pyramidal numbers and then the (k+2)gonal hyperpyramidal numbers for dimension greather than 3 (The (k,1)Pascal triangle , for the rectangular version, will generate those in its columns.) The original Pascal triangle, which is thus the (1,1)Pascal triangle, generates (both in its columns for the rectangular version and in its falling interior diagonals starting from the rightmost one, since the (1,1)Pascal triangle is symmetrical) the triangular gnomonic numbers (the natural numbers,) the triangular numbers, the tetrahedral numbers (the triangular pyramidal numbers) and then the hypertetrahedral numbers (the triangular hyperpyramidal numbers) for dimension greather than 3.
The rectangular version of the (1,k)Pascal triangle n = 0 k 1 1 k 2 1 k 3 1 k 4 1 k 5 1 k 6 1 k 7 1 k 8 1 k 9 1 k 10 1 k 11 1 k 12 1 k j = 0 1 2 3 4 5 6 7 8 9 10 11 12
In the equilateral version of the (1,k)Pascal triangle, we start with a cell (row 0) initialized to k, with the leftmost nonzero cell in the row below initialized to 1, 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 (1,k)Pascal triangle, we start with a cell (row 0) initialized to k, with the cell below it initialized to 1, 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.
The rightmost nonzero cells on each rows are therefore set to k and the leftmost nonzero cells on each rows except the first one (for n = 0) are set to 1. All the interior cells are necessarily greater than or equal to k+1. The number of cells from rows 0 to n which are equal to 1 is n. (Cf. A001477(n),) the number of cells from rows 0 to n which are equal to k is n+1 (Cf. A001477(n+1),) and the number of cells from rows 0 to n which are greater than or equal to k+1 is , the (n1)^{th} triangular number.
Recursion rule
The (1,k)Pascal triangle recursion rule is:
Formulae
Formulae in terms of binomial coefficients
where when n < 0, r < 0 or n  r < 0,^{[1]} and is cell (n, j) of Pascal's triangle.
Formulae in terms of (hyper)pyramidal numbers
 ^{[2]}
where
 ^{[2]}
and where
is the the n^{th} kstep or (k+2)gonal gnomonic number, and
is the the n^{th} (k+2)gonal number.
(1,k)Pascal triangle rows
The (1,k)Pascal triangle rows give an infinite sequence of finite sequences:
 {{}, {}, {}, {}, ...}
The concatenation of the infinite sequence of finite sequences gives the infinite sequence:
 {}
(1,k)Pascal triangle rows sums
The sums of the respective finite sequences give the infinite sequence:
 {k, k+1, 2(k+1), 4(k+1), 8(k+1), 16(k+1), 32(k+1), 64(k+1), ...}
with members given by the formula:
where:
The generating function is:
(1,k)Pascal triangle rows alternating sign sums
The alternating sign sums of the respective finite sequences give the infinite sequence:
 {k, 1k, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...}
with members given by the formula:
where:
The generating function is:
(1,k)Pascal (rectangular) triangle columns (and Chebyshev polynomials?)
Table of columns sequences
The i ^{th}, i ≥ 0, member of column j appears in row j+i.
j  sequences  OEIS
number 

0  {k, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}  
1  {}  
2  {}  
3  {}  
4  {}  
5  {}  
6  {}  
7  {}  
8  {}  
9  {}  
10  {}  
11  {}  
12  {} 
Table of columns sequences related formulae
The i ^{th}, i ≥ 0, member of column j appears in row j+i.
j  Formulae
 Generating
function for i ^{th} (i ≥ 0) member of column
 Order
of basis
 Differences
 Partial sums
 Partial sums of reciprocals
 Sum of Reciprocals^{[3]}^{[4]}


0  
1 

 
2 
 
3 
 
4  
5  
6  
7  
8  
9  
10  
11  
12 
(1,k)Pascal (rectangular) triangle falling diagonals and (k+2)gonal (hyper)pyramidal numbers
If you look at the (1,k)Pascal triangle, it seems that the convention giving the most symmetry in the triangle would be to have 1 (for the initial dot) for n = 0, so n would coincide with j. With this convention, n would indicate the number of nondegenerate subfigures of the figurate number.
The rectangular version of the (1,k)Pascal triangle
(Figurate Number Triangle)^{[5]}n = 0 k 1 1 ^{[6]} k 2 1 ^{[7]} k 3 1 ^{[2]} k 4 1 k 5 1 k 6 1 k 7 1 k 8 1 k 9 1 k 10 1 k 11 1 k 12 1 k j = 0 1 2 3 4 5 6 7 8 9 10 11 12
(1,k)Pascal (rectangular) triangle rising diagonals and ????? numbers
(1,k)Pascal triangle central elements
The central elements (for row 2m, m ≥ 0) of the (1,k)Pascal triangle give the sequence:
where A000984(n), n ≥ 0, are the central binomial coefficients, :
 {1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520, 601080390, 2333606220, 9075135300, 35345263800, ...}
They central elements are given by the formulae:
where:
is the m^{th}, m ≥ 0, Catalan number (also called Segner numbers) (Cf. A000108(m)):
 {1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, ...}
The generating function, for m ≥ 1, is:
and the generating function, for m ≥ 0, is:
where is the generating function of the Catalan numbers:
and
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., Binomial Coefficient, From MathWorldA Wolfram Web Resource.
 ↑ ^{2.0} ^{2.1} ^{2.2} Where , k ≥ 1, n ≥ 0, is the ddimensional, d ≥ 0, (k+2)gonal base (hyper)pyramidal number where, for d ≥ 2, is the number of vertices (including the apex vertices) of the polygonal base (hyper)pyramid.
 ↑ Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
 ↑ PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES.
 ↑ Weisstein, Eric W., Figurate Number Triangle, From MathWorldA Wolfram Web Resource.
 ↑ Where , k ≥ 1, n ≥ 0, is the n^{th} kstep or (k+2)gonal gnomonic number.
 ↑ Where , k ≥ 1, n ≥ 0, is the n^{th} (k+2)gonal number.
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.