A087962 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!.

0, 1, -2, 15, -220, 5025, -159606, 6593041, -338977416, 21032339985, -1539275365450, 130569297615801, -12660181105282668, 1387510663815243721, -170295099173001030606, 23224872340978381412865, -3496270002640563444940816, 577651124287028261031912609

Offset: 0

Author

Paul D. Hanna, Sep 18 2003

Keywords

• sign

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?

Examples

			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).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..282

Crossrefs

Cf. A087961.

Programs

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 *)