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

%I A216973 #16 Jul 16 2019 03:41:18
%S A216973 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,
%T A216973 42,105,140,105,42,7,0,8,56,168,280,280,168,56,8,0,9,72,252,504,630,
%U A216973 504,252,72,9,0,10,90,360,840,1260,1260,840,360,90,10,0
%N A216973 Exponential Riordan array [x*exp(x),x].
%C A216973 This is the triangle of denominators from Leibniz's harmonic triangle, A003506, augmented with a main diagonal of 0's.
%C A216973 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
%F A216973 T(n,k) = (n-k)*binomial(n,k) for 0 <= k <= n.
%F A216973 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! + ....
%F A216973 The exponential Riordan array [x*exp(x),x] factors as [x,x]*[exp(x),x] = A132440*A007318.
%F A216973 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.
%F A216973 The first column of the array Exp(t*A216973) is the sequence of idempotent polynomials, the row polynomials of A059297.
%e A216973 Triangle begins
%e A216973 .n\k.|..0.....1.....2.....3.....4.....5.....6
%e A216973 = = = = = = = = = = = = = = = = = = = = = = =
%e A216973 ..0..|..0
%e A216973 ..1..|..1.....0
%e A216973 ..2..|..2.....2.....0
%e A216973 ..3..|..3.....6.....3.....0
%e A216973 ..4..|..4....12....12.....4.....0
%e A216973 ..5..|..5....20....30....20.....5.....0
%e A216973 ..6..|..6....30....60....60....30.....6.....0
%e A216973 ...
%p A216973 A216973_row := proc(n) x*exp(x)*exp(x*t): series(%,x,n+1): n!*coeff(%,x,n):
%p A216973 seq(coeff(%,t,k), k=0..n) end:
%p A216973 for n from 0 to 10 do A216973_row(n) od; # _Peter Luschny_, Feb 03 2017
%t A216973 (* The function RiordanArray is defined in A256893. *)
%t A216973 RiordanArray[# Exp[#]&, Identity, 11, True] // Flatten (* _Jean-François Alcover_, Jul 16 2019 *)
%Y A216973 A003506, A007318, A059297, A116071, A132440, A215652.
%K A216973 nonn,easy,tabl
%O A216973 0,4
%A A216973 _Peter Bala_, Sep 21 2012