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A215652 Exponential Riordan array [exp(x*exp(-x)),x].

%I A215652 #15 Feb 13 2017 10:49:53
%S A215652 1,1,1,-1,2,1,-2,-3,3,1,9,-8,-6,4,1,-4,45,-20,-10,5,1,-95,-24,135,-40,
%T A215652 -15,6,1,414,-665,-84,315,-70,-21,7,1,49,3312,-2660,-224,630,-112,-28,
%U A215652 8,1,-10088,441,14904,-7980,-504,1134,-168,-36,9,1
%N A215652 Exponential Riordan array [exp(x*exp(-x)),x].
%C A215652 For commuting lower unitriangular matrices A and B we define A raised to the matrix power B, denoted A^^B, to be the matrix Exp(B*Log(A)). Here Exp denotes the matrix exponential and the matrix logarithm Log(A) is defined as sum {n >= 1} (-1)^(n+1)*(A-1)^n/n. Call the present triangle X and let P denote Pascal's triangle A007318. Then X solves the matrix equation X^^P = P. Equivalently, the infinite tower of matrix powers X^^(X^^(X^^(....))) equals P. Note that the infinite tower of powers P^^(P^^(P^^(...))) of the Pascal triangle equals the hyperbinomial array A088956. Thus we might view the present array as the hypobinomial triangle.
%H A215652 G. Helms, <a href="http://go.helms-net.de/math/tetdocs/PascalMatrixTetrated.pdf">Pascalmatrix tetrated</a>
%F A215652 T(n,k) = binomial(n,k)*A003725(n-k).
%F A215652 The triangle equals P^^Q, where P is Pascal's triangle and Q is the inverse of P. Column 0 equals A003725.
%F A215652 E.g.f.: exp(x*t)*exp(x*exp(-x)) = 1 + (1 + t)*x + (-1 + 2*t + t^2)*x^2/2! + (-2 - 3*t + 3*t^2 + t^3)*x^3/3! + ....
%F A215652 The infinitesimal generator for this triangle is the generalized exponential Riordan array [x*exp(-x),x], which factors as [x,x]*[exp(-x),x] = A132440*A007318^(-1). The infinitesimal generator begins
%F A215652 ..0
%F A215652 ..1....0
%F A215652 .-2....2....0
%F A215652 ..3...-6....3....0
%F A215652 .-4...12..-12....4....0
%F A215652 This is a signed version of the triangle of denominators from Leibniz's harmonic triangle - see A003506.
%e A215652 Triangle begins
%e A215652 .n\k.|....0.....1.....2.....3.....4.....5.....6.....7
%e A215652 = = = = = = = = = = = = = = = = = = = = = = = = = = =
%e A215652 ..0..|....1
%e A215652 ..1..|....1.....1
%e A215652 ..2..|...-1.....2.....1
%e A215652 ..3..|...-2....-3.....3.....1
%e A215652 ..4..|....9....-8....-6.....4.....1
%e A215652 ..5..|...-4....45...-20...-10.....5.....1
%e A215652 ..6..|..-95...-24...135...-40...-15.....6.....1
%e A215652 ..7..|..414..-665...-84...315...-70...-21.....7.....1
%e A215652 ...
%t A215652 max = 9; MapIndexed[ Take[#1, #2[[1]]]&, CoefficientList[ Series[ Exp[x*t]*Exp[x*Exp[-x]], {x, 0, max}, {t, 0, max}], {x, t}]*Range[0, max]!, 1] // Flatten (* _Jean-François Alcover_, Jan 08 2013 *)
%Y A215652 Cf. A003506, A003725 (column 0), A007318, A088956.
%K A215652 sign,easy,tabl
%O A215652 0,5
%A A215652 _Peter Bala_, Sep 11 2012