VOOZH about

URL: https://oeis.org/A381932

⇱ A381932 - OEIS

login
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.
1
1, 1, -1, 1, -5, 1, 1, -13, 1, -1, 1, -77, 89, -91, 11, 1, -29, 175, -149, 91, -1, 1, -223, 1501, -37, 391, -43, -11, 1, -481, 2821, -13943, 725, -2357, 17, 29, 1, -4609, 16099, -19481, 91313, -55649, 23137, 1727, 493, 1, -4861, 89993, -933293, 399637, -1061231, 2035739, -8189, 4897, -2711
OFFSET
1,5
COMMENTS
The main entry for this sequence of fractions is in A381931.
FORMULA
Conjecture: abs(T(n, 2)) = A064169(n - 1).
T(n, n) = numerator(A180609(n)/(n!*(n+1)!)).
EXAMPLE
Triangle T(n, k) begins:
[1] 1;
[2] 1, -1;
[3] 1, -5, 1;
[4] 1, -13, 1, -1;
[5] 1, -77, 89, -91, 11;
[6] 1, -29, 175, -149, 91, -1;
[7] 1, -223, 1501, -37, 391, -43, -11;
[8] 1, -481, 2821, -13943, 725, -2357, 17, 29;
[9] 1, -4609, 16099, -19481, 91313, -55649, 23137, 1727, 493;
.
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)
+ ... .
PROG
(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, numerator(polcoeff(p, n-k+1)));
CROSSREFS
Cf. A381931 (denominators).
Sequence in context: A299221 A300035 A391046 * A130227 A114123 A324009
KEYWORD
sign,frac,tabl
AUTHOR
Thomas Scheuerle, Mar 12 2025
STATUS
approved