A180654 - OEIS

%I #2 Mar 30 2012 18:37:22

%S 1,1,3,-13,-321,13434,103022,-60330726,4269491916,422156508320,

%T -186525936386808,22409109754552542,6675208135884604731,

%U -4757044765774305527628,1070232275818826170463982

%N E.g.f.: A(x) = Sum_{n>=0} log( Sum_{k=0..n} C(n,k)^2*x^k )^n*x^n/(n!*n^n).

%C Compare g.f. to: 1+x = Sum_{n>=0} log( (1+x)^n )^n*x^n/(n!*n^n).

%C Conjecture 1: this sequence consists entirely of integers.

%C Conjecture 2: the sequence of coefficients of [x^n/n! ] in the series:

%C . F(x,p) = Sum_{n>=0} log( Sum_{k=0..n} C(n,k)^p*x^k )^n*x^n/(n!*n^n)

%C consists entirely of integers for integer p>=1.

%C Conjecture 3: the sequence of coefficients of [x^n/n! ] in the series:

%C . G(x) = Sum_{n>=0} log( Sum_{k=0..n} C(n,k)^n*x^k )^n*x^n/(n!*n^n)

%C consists entirely of integers.

%e E.g.f.: A(x) = 1 + x + 3*x^2/2 - 13*x^3/3! - 321*x^4/4! + 13434*x^5/5! +...

%e 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) +...

%o (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)}

%K sign

%O 0,3

%A _Paul D. Hanna_, Sep 14 2010