A307231 - OEIS

%I #9 May 30 2019 22:54:11

%S 1,1,1,2,3,6,14,31,74,176,425,1055,2637,6671,16995,43507,112102,

%T 290347,755836,1976717,5189664,13673314,36139543,95795807,254610593,

%U 678385919,1811625931,4848177910,12999914523,34921821516,93971264941,253272232362,683646029385,1847935127768,5001703392561,13554768152442,36777239050602,99896764310264,271633640605450,739353335433377,2014350935223715,5493039157088226,14992221809540249

%N G.f.: A(x) = exp( Sum_{n>=1} x^n/n * (1 + x^n*A(x)^(2*n)) / (1 + x^n*A(x)^n) ).

%H Paul D. Hanna, <a href="/A307231/b307231.txt">Table of n, a(n) for n = 0..300</a>

%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 14*x^6 + 31*x^7 + 74*x^8 + 176*x^9 + 425*x^10 + 1055*x^11 + 2637*x^12 + 6671*x^13 + 16995*x^14 + 43507*x^15 + ...

%e such that

%e log(A(x)) = x*(1 + x*A(x)^2)/(1 + x*A(x)) + x^2/2*(1 + x^2*A(x)^4)/(1 + x^2*A(x)^2) + x^3/3*(1 + x^3*A(x)^6)/(1 + x^3*A(x)^3) + x^4/4*(1 + x^4*A(x)^8)/(1 + x^4*A(x)^4) + x^5/5*(1 + x^5*A(x)^10)/(1 + x^5*A(x)^5) + ...

%e explicitly,

%e log(A(x)) = x + x^2/2 + 4*x^3/3 + 5*x^4/4 + 16*x^5/5 + 46*x^6/6 + 113*x^7/7 + 317*x^8/8 + 823*x^9/9 + 2206*x^10/10 + 6051*x^11/11 + 16418*x^12/12 + 45007*x^13/13 + 123033*x^14/14 + 336244*x^15/15 + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=Vec( exp( sum(m=1,#A+1, x^m/m * (1 + x^m*Ser(A)^(2*m))/(1 + x^m*Ser(A)^m) ) )) );A[n+1]}

%o for(n=0,40,print1(a(n),", "))

%K nonn

%O 0,4

%A _Paul D. Hanna_, Mar 29 2019