cp's OEIS Frontend

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.

%I A381931 #23 Mar 18 2025 20:25:39
%S A381931 2,4,12,8,48,48,16,144,24,180,32,1152,1728,5760,8640,64,640,3456,5760,
%T A381931 17280,6720,128,7680,34560,1152,34560,32256,241920,256,26880,82944,
%U A381931 414720,41472,580608,107520,1451520,512,430080,645120,622080,4147200,6967296,21772800,87091200,43545600
%N 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.
%C A381931 This is the main entry for this sequence of fractions.
%C A381931 Convergence and analytic continuation of this series representation are interesting research topics with many unsolved problems and open questions.
%C A381931 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)!.
%H A381931 Gottfried Helms, <a href="https://go.helms-net.de/math/tetdocs/CoefficientsForUTetration.htm">Coefficients for fractional iterates exp(x)-1</a>
%H A381931 MathOverflow, <a href="https://mathoverflow.net/questions/4347/ffx-expx-1-and-other-functions-just-in-the-middle-between-linear-and">f(f(x)) = exp(x)-1 and other functions "just in the middle" between linear and exponential</a>
%F A381931 T(n, 1) = 2^n.
%F A381931 T(n, n) = denominator(A180609(n)/(n!*(n+1)!)).
%e A381931 Triangle T(n, k) begins:
%e A381931 [1]  2;
%e A381931 [2]  4,   12;
%e A381931 [3]  8,   48,     48;
%e A381931 [4]  16,  144,    24,     180;
%e A381931 [5]  32,  1152,   1728,   5760,   8640;
%e A381931 [6]  64,  640,    3456,   5760,   17280,   6720;
%e A381931 [7]  128, 7680,   34560,  1152,   34560,   32256,   241920;
%e A381931 [8]  256, 26880,  82944,  414720, 41472,   580608,  107520,   1451520;
%e A381931 [9]  512, 430080, 645120, 622080, 4147200, 6967296, 21772800, 87091200, 43545600;
%e A381931 .
%e A381931 F^{r}(x) = x
%e A381931 + x^2*1/2*r
%e A381931 + x^3*(1/4*r^2 - 1/12*r)
%e A381931 + x^4*(1/8*r^3 - 5/48*r^2 + 1/48*r)
%e A381931 + x^5*(1/16*r^4 - 13/144*r^3 + 1/24*r^2 - 1/180*r)
%e A381931 + x^6*(1/32*r^5 - 77/1152*r^4 + 89/1728*r^3 - 91/5760*r^2 + 11/8640*r)
%e A381931 + ... .
%o A381931 (PARI)
%o A381931 c(k, n) = {my(f=x); for(m=1, k, f=subst(f, x, exp(x)-1)); polcoeff(f+O(x^(n+1)), n)}
%o A381931 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)));
%Y A381931 Cf. A381932 (numerators).
%Y A381931 Cf. A052123, A052122, A052104, A052105, A144150, A180609, A184011.
%K A381931 nonn,frac,tabl
%O A381931 1,1
%A A381931 _Thomas Scheuerle_, Mar 10 2025