A381931 Triangular array T(n, k) read by rows: denominators of the coefficients for the iterated exponential F^{r}(x) = x + Sum_{n>=1} x^(n+1)*Sum_{k=1..n} r^(n+1-k)*A381932(n, k)/T(n, k) with F^{1}(x) = exp(x)-1 and F^{2}(x) = exp(exp(x)-1)-1.

2, 4, 12, 8, 48, 48, 16, 144, 24, 180, 32, 1152, 1728, 5760, 8640, 64, 640, 3456, 5760, 17280, 6720, 128, 7680, 34560, 1152, 34560, 32256, 241920, 256, 26880, 82944, 414720, 41472, 580608, 107520, 1451520, 512, 430080, 645120, 622080, 4147200, 6967296, 21772800, 87091200, 43545600

Offset: 1

Author

Thomas Scheuerle, Mar 10 2025

Keywords

• nonn
• frac
• tabl

Comments

This is the main entry for this sequence of fractions.

Convergence and analytic continuation of this series representation are interesting research topics with many unsolved problems and open questions.

Evaluating the polynomial of row n P(x) = Sum_{k=1..n} x^(n+1-k)*A381932(n, k)/T(n, k) gives A144150(n+1, x-1)/(n+1)!.

Examples

			Triangle T(n, k) begins:
[1]  2;
[2]  4,   12;
[3]  8,   48,     48;
[4]  16,  144,    24,     180;
[5]  32,  1152,   1728,   5760,   8640;
[6]  64,  640,    3456,   5760,   17280,   6720;
[7]  128, 7680,   34560,  1152,   34560,   32256,   241920;
[8]  256, 26880,  82944,  414720, 41472,   580608,  107520,   1451520;
[9]  512, 430080, 645120, 622080, 4147200, 6967296, 21772800, 87091200, 43545600;
.
F^{r}(x) = x
+ x^2*1/2*r
+ x^3*(1/4*r^2 - 1/12*r)
+ x^4*(1/8*r^3 - 5/48*r^2 + 1/48*r)
+ x^5*(1/16*r^4 - 13/144*r^3 + 1/24*r^2 - 1/180*r)
+ x^6*(1/32*r^5 - 77/1152*r^4 + 89/1728*r^3 - 91/5760*r^2 + 11/8640*r)
+ ... .
		

Links

• Gottfried Helms, Coefficients for fractional iterates exp(x)-1
• MathOverflow, f(f(x)) = exp(x)-1 and other functions "just in the middle" between linear and exponential

Crossrefs

Cf. A381932 (numerators).

Cf. A052123, A052122, A052104, A052105, A144150, A180609, A184011.

Programs

PARI

c(k, n) = {my(f=x); for(m=1, k, f=subst(f, x, exp(x)-1)); polcoeff(f+O(x^(n+1)), n)}
row(n) = my(p=polinterpolate(vector(2*(n+1), k, k-1), vector(2*(n+1), k, c(k-1, n+1)))); vector(n, k, denominator(polcoeff(p, n-k+1)));

Formula

T(n, 1) = 2^n.

T(n, n) = denominator(A180609(n)/(n!*(n+1)!)).