A087962 - OEIS

%I #14 May 21 2022 08:30:03

%S 0,1,-2,15,-220,5025,-159606,6593041,-338977416,21032339985,

%T -1539275365450,130569297615801,-12660181105282668,

%U 1387510663815243721,-170295099173001030606,23224872340978381412865,-3496270002640563444940816,577651124287028261031912609

%N Satisfies Sum_{n>=0} a(n)*x^n/n! = log(f(x)) = series reversion of x*f(x), where f(x*f(x)) = exp(x) and f(x) = Sum_{n>=0} A087961(n)*x^n/n!.

%C 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?

%H Alois P. Heinz, <a href="/A087962/b087962.txt">Table of n, a(n) for n = 0..282</a>

%e f(x) = 1 +1x -1x^2/2! +10x^3/3! -159x^4/4! +3816x^5/5! -125375x^6/6! +-...

%e where f(xf(x)) = exp(x).

%p b:= proc(n, k) option remember; `if`(n=0, 1/k, add(k*

%p       b(j-1, j)*j*b(n-j, k)*binomial(n-1, j-1), j=1..n))

%p     end:

%p a:= n-> -b(n-1, n)*n*(-1)^n:

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Aug 21 2019

%t b[n_, k_] := b[n, k] = If[n == 0, 1/k, Sum[k*

%t      b[j-1, j]*j*b[n-j, k]*Binomial[n-1, j-1], {j, 1, n}]];

%t a[n_] := -b[n-1, n]*n*(-1)^n;

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 21 2022, after _Alois P. Heinz_ *)

%Y Cf. A087961.

%K sign

%O 0,3

%A _Paul D. Hanna_, Sep 18 2003