OFFSET
0,3
COMMENTS
Compare g.f. to: 1+x = Sum_{n>=0} log( (1+x)^n )^n*x^n/(n!*n^n).
Conjecture 1: this sequence consists entirely of integers.
Conjecture 2: the sequence of coefficients of [x^n/n! ] in the series:
. F(x,p) = Sum_{n>=0} log( Sum_{k=0..n} C(n,k)^p*x^k )^n*x^n/(n!*n^n)
consists entirely of integers for integer p>=1.
Conjecture 3: the sequence of coefficients of [x^n/n! ] in the series:
. G(x) = Sum_{n>=0} log( Sum_{k=0..n} C(n,k)^n*x^k )^n*x^n/(n!*n^n)
consists entirely of integers.
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2 - 13*x^3/3! - 321*x^4/4! + 13434*x^5/5! +...
A(x) = 1 + log(1+x) + log(1+4*x+x^2)^2/(2!*2^2) + log(1+9*x+9*x^2+x^3)^3/(3!*3^3) + log(1+16*x+36*x^2+16*x^3+x^4)^4/(4!*4^4) + log(1+25*x+100*x^2+100*x^3+25*x^4+x^5)^5/(5!*5^5) +...
PROG
(PARI) {a(n)=local(A=1+sum(m=1, n, log(sum(k=0, m, binomial(m, k)^2*x^k)+x*O(x^n))^m/m^m/m!)); n!*polcoeff(A, n)}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Sep 14 2010
STATUS
approved