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

0,3

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

References

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

Links

Table of n, a(n) for n=0..54.

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.

Example

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.

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);

Prog

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

Crossrefs

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

Sequence in context: A151668 A086151 A099040 * A136717 A261685 A136716

Adjacent sequences: A185293 A185294 A185295 * A185297 A185298 A185299

Keyword

nonn,easy,tabl

Author

Peter Bala, Feb 20 2011

Status

approved