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A216973 Exponential Riordan array [x*exp(x),x]. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

LINKS

Table of n, a(n) for n=0..65.

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.

EXAMPLE

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

...

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-Fran├žois Alcover, Jul 16 2019 *)

CROSSREFS

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

Sequence in context: A098816 A214639 A319495 * A061314 A193383 A218033

Adjacent sequences:  A216970 A216971 A216972 * A216974 A216975 A216976

KEYWORD

nonn,easy,tabl

AUTHOR

Peter Bala, Sep 21 2012

STATUS

approved

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Last modified November 21 00:09 EST 2019. Contains 329348 sequences. (Running on oeis4.)