OFFSET
0,3
COMMENTS
This is the series reversion of xf(x) where f(xf(x))=exp(x), exp(xf(x))=f(xf(x)*exp(x)), f(log(x)*f(log(x)))=x and f(x)=sum(n>=0, A087961(n)*x^n/n!). Are these series convergent anywhere besides at x=0?
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..282
EXAMPLE
f(x) = 1 +1x -1x^2/2! +10x^3/3! -159x^4/4! +3816x^5/5! -125375x^6/6! +-...
where f(xf(x)) = exp(x).
MAPLE
b:= proc(n, k) option remember; `if`(n=0, 1/k, add(k*
b(j-1, j)*j*b(n-j, k)*binomial(n-1, j-1), j=1..n))
end:
a:= n-> -b(n-1, n)*n*(-1)^n:
seq(a(n), n=0..20); # Alois P. Heinz, Aug 21 2019
MATHEMATICA
b[n_, k_] := b[n, k] = If[n == 0, 1/k, Sum[k*
b[j-1, j]*j*b[n-j, k]*Binomial[n-1, j-1], {j, 1, n}]];
a[n_] := -b[n-1, n]*n*(-1)^n;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 21 2022, after Alois P. Heinz *)
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Sep 18 2003
STATUS
approved