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A381932 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)*T(n, k)/A381931(n, k) with F^{1}(x) = exp(x)-1 and F^{2}(x) = exp(exp(x)-1)-1.

%I A381932 #9 Mar 18 2025 20:25:49
%S A381932 1,1,-1,1,-5,1,1,-13,1,-1,1,-77,89,-91,11,1,-29,175,-149,91,-1,1,-223,
%T A381932 1501,-37,391,-43,-11,1,-481,2821,-13943,725,-2357,17,29,1,-4609,
%U A381932 16099,-19481,91313,-55649,23137,1727,493,1,-4861,89993,-933293,399637,-1061231,2035739,-8189,4897,-2711
%N A381932 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)*T(n, k)/A381931(n, k) with F^{1}(x) = exp(x)-1 and F^{2}(x) = exp(exp(x)-1)-1.
%C A381932 The main entry for this sequence of fractions is in A381931.
%F A381932 Conjecture: abs(T(n, 2)) = A064169(n - 1).
%F A381932 T(n, n) = numerator(A180609(n)/(n!*(n+1)!)).
%e A381932 Triangle T(n, k) begins:
%e A381932 [1]  1;
%e A381932 [2]  1,    -1;
%e A381932 [3]  1,    -5,     1;
%e A381932 [4]  1,   -13,     1,     -1;
%e A381932 [5]  1,   -77,    89,    -91,    11;
%e A381932 [6]  1,   -29,   175,   -149,    91,     -1;
%e A381932 [7]  1,  -223,  1501,    -37,   391,    -43,   -11;
%e A381932 [8]  1,  -481,  2821, -13943,   725,  -2357,    17,   29;
%e A381932 [9]  1, -4609, 16099, -19481, 91313, -55649, 23137, 1727, 493;
%e A381932 .
%e A381932 F^{r}(x) = x
%e A381932 + x^2*1/2*r
%e A381932 + x^3*(1/4*r^2 - 1/12*r)
%e A381932 + x^4*(1/8*r^3 - 5/48*r^2 + 1/48*r)
%e A381932 + x^5*(1/16*r^4 - 13/144*r^3 + 1/24*r^2 - 1/180*r)
%e A381932 + x^6*(1/32*r^5 - 77/1152*r^4 + 89/1728*r^3 - 91/5760*r^2 + 11/8640*r)
%e A381932 + ... .
%o A381932 (PARI)
%o A381932 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 A381932 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, numerator(polcoeff(p, n-k+1)));
%Y A381932 Cf. A381931 (denominators).
%Y A381932 Cf. A052123, A052122, A052104, A052105
%Y A381932 Cf. A064169, A144150, A180609, A184011.
%K A381932 sign,frac,tabl
%O A381932 1,5
%A A381932 _Thomas Scheuerle_, Mar 12 2025