A186432 Triangle associated with the set S of squares {0,1,4,9,16,...}.

1, 1, 1, 1, 12, 1, 1, 30, 30, 1, 1, 56, 140, 56, 1, 1, 90, 420, 420, 90, 1, 1, 132, 990, 1848, 990, 132, 1, 1, 182, 2002, 6006, 6006, 2002, 182, 1, 1, 240, 3640, 16016, 25740, 16016, 3640, 240, 1, 1, 306, 6120, 37128, 87516, 87516, 37128, 6120, 306, 1, 1, 380, 9690, 77520, 251940, 369512, 251940, 77520, 9690, 380, 1

Offset: 0

Author

Peter Bala, Feb 22 2011

Keywords

• nonn
• easy
• tabl

Comments

Given a subset S of the integers Z, Bhargava [1] has shown how to associate with S a generalized factorial function, denoted n!_S, sharing many properties of the classical factorial function n! (which corresponds to the choice S = Z). In particular, he shows that the generalized binomial coefficients n!_S/(k!_S*(n-k)!_S) are always integral for any choice of S. Here we take S = {0,1,4,9,16,...}, the set of squares.

The associated generalized factorial function n!_S is given by the formula

n!S = Product{k=0..n} (n^2 - k^2), with the convention 0!S = 1. This should be compared with n! = Product{k=0..n} (n - k).

For n >= 1, n!_S = (2*n)!/2 = A002674(n).

Compare this triangle with A086645 and also A186430 - the generalized binomial coefficients for the set S of prime numbers {2,3,5,7,11,...}.

Examples

			Triangle begins
n/k.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....1
.2..|..1....12.....1
.3..|..1....30....30.....1
.4..|..1....56...140....56.....1
.5..|..1....90...420...420....90.....1
.6..|..1...132...990..1848...990...132.....1
.7..|..1...182..2002..6006..6006..2002...182.....1
...
		

Links

• M. Bhargava, The factorial function and generalizations, Amer. Math. Monthly, 107 (2000), 783-799.

Crossrefs

Cf. A002114, A086645, A186430, A186433 (inverse).

Programs

Mathematica

Table[2 Binomial[2 n, 2 k] - Boole[Or[k == 0, k == n]], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 23 2017 *)

Formula

TABLE ENTRIES

T(n,k) = n!_S/(k!_S*(n-k)!_S),

which simplifies to

T(n,k) = 2*binomial(2*n,2*k) for 1 <= k < n,

with boundary conditions T(n,0) = 1 and T(n,n) = 1 for n >= 0.

RELATIONS WITH OTHER SEQUENCES

Denote this triangle by T. The first column of the inverse T^-1 (see A186433) begins [1, -1, 11, -301, 15371, ...] and, apart from the initial 1, is a signed version of the Glaisher's H' numbers A002114.

The first column of (1/2)*T^2 begins [1/2, 1, 7, 31, 127, ...] and, apart from the initial term, equals A000225(2*n-1), counting the preferential arrangements on (2*n - 1) labeled elements having less than or equal to two ranks.

The first column of (1/3)*T^3 begins [1/3, 1, 13, 181, 1933, ...] and, apart from the initial term, is A101052(2*n-1), which gives the number of preferential arrangements on (2*n-1) labeled elements having less than or equal to three ranks.