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

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, -15, 6, 1, 414, -665, -84, 315, -70, -21, 7, 1, 49, 3312, -2660, -224, 630, -112, -28, 8, 1, -10088, 441, 14904, -7980, -504, 1134, -168, -36, 9, 1

Offset: 0

Author

Peter Bala, Sep 11 2012

Keywords

• sign
• easy
• tabl

Comments

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.

Examples

			Triangle begins
.n\k.|....0.....1.....2.....3.....4.....5.....6.....7
= = = = = = = = = = = = = = = = = = = = = = = = = = =
..0..|....1
..1..|....1.....1
..2..|...-1.....2.....1
..3..|...-2....-3.....3.....1
..4..|....9....-8....-6.....4.....1
..5..|...-4....45...-20...-10.....5.....1
..6..|..-95...-24...135...-40...-15.....6.....1
..7..|..414..-665...-84...315...-70...-21.....7.....1
...
		

Links

• G. Helms, Pascalmatrix tetrated

Crossrefs

Cf. A003506, A003725 (column 0), A007318, A088956.

Programs

Mathematica

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 *)

Formula

T(n,k) = binomial(n,k)*A003725(n-k).

The triangle equals P^^Q, where P is Pascal's triangle and Q is the inverse of P. Column 0 equals A003725.

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! + ....

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

..0

..1....0

.-2....2....0

..3...-6....3....0

.-4...12..-12....4....0

This is a signed version of the triangle of denominators from Leibniz's harmonic triangle - see A003506.