A216973 Exponential Riordan array [x*exp(x),x].

0, 1, 0, 2, 2, 0, 3, 6, 3, 0, 4, 12, 12, 4, 0, 5, 20, 30, 20, 5, 0, 6, 30, 60, 60, 30, 6, 0, 7, 42, 105, 140, 105, 42, 7, 0, 8, 56, 168, 280, 280, 168, 56, 8, 0, 9, 72, 252, 504, 630, 504, 252, 72, 9, 0, 10, 90, 360, 840, 1260, 1260, 840, 360, 90, 10, 0

Offset: 0

Author

Peter Bala, Sep 21 2012

Keywords

• nonn
• easy
• tabl

Comments

This is the triangle of denominators from Leibniz's harmonic triangle, A003506, augmented with a main diagonal of 0's.

Note, the usual definition of the exponential Riordan array [f(x), x*g(x)] associated with a pair of power series f(x) and g(x) requires f(0) to be nonzero. Here we don't make this assumption. - Peter Bala, Feb 13 2017

Examples

			Triangle begins
.n\k.|..0.....1.....2.....3.....4.....5.....6
= = = = = = = = = = = = = = = = = = = = = = =
..0..|..0
..1..|..1.....0
..2..|..2.....2.....0
..3..|..3.....6.....3.....0
..4..|..4....12....12.....4.....0
..5..|..5....20....30....20.....5.....0
..6..|..6....30....60....60....30.....6.....0
...
		

Crossrefs

A003506, A007318, A059297, A116071, A132440, A215652.

Programs

Maple

A216973_row := proc(n) x*exp(x)*exp(x*t): series(%,x,n+1): n!*coeff(%,x,n):
seq(coeff(%,t,k), k=0..n) end:
for n from 0 to 10 do A216973_row(n) od; # Peter Luschny, Feb 03 2017

Mathematica

(* The function RiordanArray is defined in A256893. *)
RiordanArray[# Exp[#]&, Identity, 11, True] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Formula

T(n,k) = (n-k)*binomial(n,k) for 0 <= k <= n.

E.g.f.: x*exp(x)*exp(x*t) = 1 + x + (2 + 2*t)*x^2/2! + (3 + 6*t + 3*t^2)*x^3/3! + ....

The exponential Riordan array [x*exp(x),x] factors as [x,x]*[exp(x),x] = A132440*A007318.

This array is the infinitesimal generator for A116071; that is, Exp(A216973) = A116071, where Exp denotes the matrix exponential. A signed version of the array is the infinitesimal generator for A215652.

The first column of the array Exp(t*A216973) is the sequence of idempotent polynomials, the row polynomials of A059297.