A294115 G.f.: exp( Sum_{n>=1} L(n) * x^n/n ), where Sum_{n>=1} L(n) * x^n/n! = Series_Reversion( log(1 + x/A(x)) ).

1, 1, 2, 10, 143, 5959, 904224, 696895088, 3563009122225, 144004257475683137, 52273888783668336094726, 189699379891906830471022186526, 7572226826806850232281722700245568807, 3627110408773444347271222282038547230122245455, 22586092882428159778440302586299616247303225297287979548, 1969016989037466758104728399066094312610056241493227691736998060636, 2574833047387344521023398134994106823445574761658761070132072595536874966252691

Offset: 0

Author

Paul D. Hanna, Oct 22 2017

Keywords

• nonn

Comments

This sequence is motivated by the following conjectures:

(C1) Given integer series G(x) such that G(0) = G'(0) = 1, define L(n) by

Sum_{n>=1} L(n) * x^n/n! = Series_Reversion( log(G(x)) )

then exp( Sum_{n>=1} L(n) * x^n/n ) is also an integer series;

(C2) Given G(x) = 1 + x*G(x)^m, define L(n) by

Sum_{n>=1} L(n) * x^n/n! = Series_Reversion( log(G(x)) )

then exp( Sum_{n>=1} L(n) * x^n/n ) = (1 + m*x)/(1 + (m-1)*x).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 143*x^4 + 5959*x^5 + 904224*x^6 + 696895088*x^7 + 3563009122225*x^8 + 144004257475683137*x^9 +...
The logarithm of the g.f. begins
log(A(x)) = x + 3*x^2/2 + 25*x^3/3 + 531*x^4/4 + 29041*x^5/5 + 5388603*x^6/6 + 4871887945*x^7/7 + 28498490189571*x^8/8 + 1296006243863566561*x^9/9 +...+ L(n)*x^n/n +...
such that the same coefficients L(n) are also found in
Series_Reversion( log(1 + x/A(x)) ) = x + 3*x^2/2! + 25*x^3/3! + 531*x^4/4! + 29041*x^5/5! + 5388603*x^6/6! + 4871887945*x^7/7! + 28498490189571*x^8/8! +...+ L(n)*x^n/n! +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A293379.

Programs

PARI

{a(n) = my(A=1+x, L); for(i=0,n, L = x*serlaplace( 1/x*serreverse( log(1 + x/A +O(x^(n+2))) ) ); A = exp(L);); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))