

A215652


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


5



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
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OFFSET

0,5


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)*(A1)^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.


LINKS

Table of n, a(n) for n=0..54.
G. Helms, Pascalmatrix tetrated


FORMULA

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


EXAMPLE

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
...


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 (* JeanFrançois Alcover, Jan 08 2013 *)


CROSSREFS

Cf. A003506, A003725 (column 0), A007318, A088956.
Sequence in context: A109449 A129570 A238385 * A305715 A165014 A058063
Adjacent sequences: A215649 A215650 A215651 * A215653 A215654 A215655


KEYWORD

sign,easy,tabl


AUTHOR

Peter Bala, Sep 11 2012


STATUS

approved



