A180654 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).

1, 1, 3, -13, -321, 13434, 103022, -60330726, 4269491916, 422156508320, -186525936386808, 22409109754552542, 6675208135884604731, -4757044765774305527628, 1070232275818826170463982

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.

Links

Table of n, a(n) for n=0..14.

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

Sequence in context: A045748 A113526 A113612 * A260576 A156358 A066266

Adjacent sequences: A180651 A180652 A180653 * A180655 A180656 A180657

Keyword

sign

Author

Paul D. Hanna, Sep 14 2010

Status

approved