A185296 Triangle of connection constants between the falling factorials (x)(n) and (2*x)(n).

1, 0, 2, 0, 2, 4, 0, 0, 12, 8, 0, 0, 12, 48, 16, 0, 0, 0, 120, 160, 32, 0, 0, 0, 120, 720, 480, 64, 0, 0, 0, 0, 1680, 3360, 1344, 128, 0, 0, 0, 0, 1680, 13440, 13440, 3584, 256, 0, 0, 0, 0, 0, 30240, 80640, 48384, 9216, 512

Offset: 0

Author

Peter Bala, Feb 20 2011

Keywords

• nonn
• easy
• tabl

Comments

The falling factorial polynomials (x)_n := x*(x-1)*...*(x-n+1), n = 0,1,2,..., form a basis for the space of polynomials. Hence the polynomial (2*x)_n may be expressed as a linear combination of x_0, x_1,...,x_n; the coefficients in the expansion form the n-th row of the table. Some examples are given below.

This triangle is connected to two families of orthogonal polynomials, the Hermite polynomials H(n,x) A060821, and the Bessel polynomials y(n,x)A001498. The first few Hermite polynomials are

... H(0,x) = 1

... H(1,x) = 2*x

... H(2,x) = -2+4*x^2

... H(3,x) = -12*x+8*x^3

... H(4,x) = 12-48*x^2+16*x^4.

The unsigned coefficients of H(n,x) give the nonzero entries of the n-th row of the triangle.

The Bessel polynomials y(n,x) begin

... y(0,x) = 1

... y(1,x) = 1+x

... y(2,x) = 1+3*x+3*x^2

... y(3,x) = 1+6*x+15*x^2+15*x^3.

The entries in the n-th column of this triangle are the coefficients of the scaled Bessel polynomials 2^n*y(n,x).

Also the Bell transform of g(n) = 2 if n<2 else 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

Examples

			Triangle begins
n\k|...0.....1.....2.....3.....4.....5.....6
============================================
0..|...1
1..|...0.....2
2..|...0.....2.....4
3..|...0.....0....12.....8
4..|...0.....0....12....48....16
5..|...0.....0.....0...120...160....32
6..|...0.....0.....0...120...720...480....64
..
Row 3:
(2*x)_3 = (2*x)*(2*x-1)*(2*x-2) = 8*x*(x-1)*(x-2) + 12*x*(x-1).
Row 4:
(2*x)_4 = (2*x)*(2*x-1)*(2*x-2)*(2*x-3) = 16*x*(x-1)*(x-2)*(x-3) +
48*x*(x-1)*(x-2)+ 12*x*(x-1).
Examples of recurrence relation
T(4,4) = 5*T(3,4) + 2*T(3,3) = 5*0 + 2*8 = 16;
T(5,4) = 4*T(4,4) + 2*T(4,3) = 4*16 + 2*48 = 160;
T(6,4) = 3*T(5,4) + 2*T(5,3) = 3*160 + 2*120 = 720;
T(7,4) = 2*T(6,4) + 2*T(6,3) = 2*720 + 2*120 = 1680.
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, page 158, exercise 7.

Crossrefs

Cf. A000898 (row sums), A001498, A001515, A059343, A060821.

Programs

Maple

T := (n,k) -> (n!/k!)*binomial(k,n-k)*2^(2*k-n):
seq(seq(T(n, k), k=0..n), n=0..9);

Sage

# uses[bell_matrix from A264428]
bell_matrix(lambda n: 2 if n<2 else 0, 12) # Peter Luschny, Jan 19 2016

Formula

Defining relation: 2*x*(2*x-1)*...*(2*x-n+1) = sum {k=0..n} T(n, k)*x*(x-1)*...*(x-k+1)

Explicit formula: T(n,k) = (n!/k!)*binomial(k,n-k)*2^(2*k-n). [As defined by Comtet (see reference).]

Recurrence relation: T(n,k) = (2*k-n+1)*T(n-1,k)+2*T(n-1,k-1).

E.g.f.: exp(x*(t^2+2*t)) = 1 + (2*x)*t + (2*x+4*x^2)*t^2/2! + (12*x^2+8*x^3)*t^3/3! + ...

O.g.f. for m-th diagonal (starting at main diagonal m = 0): (2*m)!/m!*x^m/(1-2*x)^(2*m+1).

The triangle is the matrix product [2^k*s(n,k)]n,k>=0 * ([s(n,k)]n,k>=0)^(-1),

where s(n,k) denotes the signed Stirling number of the first kind.

Row sums are [1,2,6,20,76,...] = A000898.

Column sums are [1,4,28,296,...] = [2^n*A001515(n)] n>=0.