OFFSET
0,3
COMMENTS
Compare g.f. to the trivial identity: G(x) = exp(Sum_{n>=1} G(-x^n)*x^n/n) where G(x) = 1+x.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1347
FORMULA
Equals the Euler transformation of the coefficients in A(-x)^3, where A(x) is the g.f. of this sequence.
EXAMPLE
G.f.: A(x) = 1 + x - 2*x^2 - 5*x^3 + 24*x^4 + 81*x^5 - 439*x^6 +...
where
log(A(x)) = A(-x)^3*x + A(-x^2)^3*x^2/2 + A(-x^3)^3*x^3/3 + A(-x^4)^3*x^4/4 +...
The coefficients in A(-x)^3 begin:
[1,-3,-3,26,48,-444,-920,9126,19587,-204214,-449496,4841001,...]
and the g.f. may be expressed by the Euler product:
A(x) = 1/((1-x)^1*(1-x^2)^-3*(1-x^3)^-3*(1-x^4)^26*(1-x^5)^48*(1-x^6)^-444*(1-x^7)^-920*(1-x^8)^9126*...).
MAPLE
b:= proc(n) option remember; add(a(i)*a(n-i), i=0..n) end:
g:= proc(n) option remember; (-1)^n*add(b(i)*a(n-i), i=0..n) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(
d*g(d-1), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..30); # Alois P. Heinz, Jan 24 2017
MATHEMATICA
b[n_] := b[n] = Sum[a[i]*a[n-i], {i, 0, n}];
g[n_] := g[n] = (-1)^n*Sum[b[i]*a[n-i], {i, 0, n}];
a[n_] := a[n] = If[n==0, 1, Sum[DivisorSum[j, #*g[#-1]&]*a[n-j], {j, 1, n}]/n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 24 2017, after Alois P. Heinz *)
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, subst(A^3, x, -x^m)*x^m/m)+x*O(x^n))); polcoeff(A, n)}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Nov 17 2011
STATUS
approved