A180563 - OEIS

%I #11 Oct 28 2017 08:38:06

%S 1,-3,19,-207,3331,-71223,1890379,-59652687,2175761971,-89953773543,

%T 4155502117339,-212122704251967,11857607972675011,-720435277883199063,

%U 47273215180877201899,-3331797538738820992047,251025685429022007354451,-20133640365773761748643783,1712740622904757368673592059

%N E.g.f. A(x) satisfies: P(A(x)) = exp(x) where P(x) = Product_{n>=1} 1/(1-x^n), the partition function.

%C Unsigned version is A294330.

%H Paul D. Hanna, <a href="/A180563/b180563.txt">Table of n, a(n) for n = 1..260</a>

%F E.g.f.: A(x) = Series_Reversion( log(P(x)) ) where P(x) = Product_{n>=1} 1/(1-x^n).

%F From _Paul D. Hanna_, Oct 28 2017 (Start):

%F E.g.f. A(x) satisfies:

%F (1) Sum_{n>=1} sigma(n) * A(x)^n / n = x.

%F (2) Product_{n>=1} (1 - A(x)^n) = exp(-x).

%F (3) Sum_{n>=0} (-1)^n * (2*n+1) * A(x)^(n*(n+1)/2) = exp(-3*x). (End)

%F Logarithmic derivative of A294332. - _Paul D. Hanna_, Oct 28 2017

%e E.g.f.: A(x) = x - 3*x^2/2! + 19*x^3/3! - 207*x^4/4! + 3331*x^5/5! - 71223*x^6/6! + 1890379*x^7/7! - 59652687*x^8/8! + 2175761971*x^9/9! - 89953773543*x^10/10! +...

%e such that  A( log(P(x)) ) = x, where:

%e log(P(x)) = x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 6*x^5/5 +...+ sigma(n)*x^n/n +...

%e and P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + 56*x^11 + 77*x^12 +...+ A000041(n)*x^n +...

%e ALTERNATE GENERATING FUNCTION.

%e L.g.f.: L(x) = x - 3*x^2/2 + 19*x^3/3 - 207*x^4/4 + 3331*x^5/5 - 71223*x^6/6 + 1890379*x^7/7 - 59652687*x^8/8 + 2175761971*x^9/9 - 89953773543*x^10/10 +...

%e such that

%e exp(L(x)) = 1 + x - x^2 + 5*x^3 - 45*x^4 + 609*x^5 - 11141*x^6 + 257281*x^7 - 7170355*x^8 + 233936995*x^9 - 8744103079*x^10 +...+ A294332(n)*x^n +...

%o (PARI) {a(n) = local( LogPx = sum(m=1,n, sigma(m) * x^m/m ) +x*O(x^n) ); n!*polcoeff( serreverse(LogPx), n)}

%o for(n=1,20, print1(a(n),", "))

%Y Cf. A000041 (partitions), A000203 (sigma), A294332, A294330.

%K sign

%O 1,2

%A _Paul D. Hanna_, Sep 09 2010