A324617 - OEIS

%I #6 Mar 12 2019 02:08:45

%S 1,-1,1,-6,10,-27,28,-107,502,-1996,-1015,39035,-76739,-1078632,

%T 7222569,9644362,-337421969,1171731119,9909483512,-109536156966,

%U 74836320374,5651749289781,-37674051339344,-117589711277053,3186640549115616,-12979461559921647,-138543759546567508,1942263572054253138,-3322718404632175707,-132968516893238601191,1307791482651889603081,1344751233503556511150

%N G.f. A(x) satisfies: 1 + x = Sum_{n>=0} x^n*(A(x)^n + 1)^n/(1 + x*A(x)^n)^(n+1).

%H Paul D. Hanna, <a href="/A324617/b324617.txt">Table of n, a(n) for n = 0..200</a>

%F G.f. A(x) satisfies:

%F (1) 1 + x = Sum_{n>=0} x^n*(A(x)^n + 1)^n/(1 + x*A(x)^n)^(n+1).

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

%e G.f.: A(x) = 1 - x + x^2 - 6*x^3 + 10*x^4 - 27*x^5 + 28*x^6 - 107*x^7 + 502*x^8 - 1996*x^9 - 1015*x^10 + 39035*x^11 - 76739*x^12 - 1078632*x^13 + ...

%e such that

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

%e also

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

%o (PARI) {a(n) = my(A=[1]); for(i=0,n, A = concat(A,0);

%o A[#A] = -polcoeff( sum(n=0,#A+1, x^n*(Ser(A)^n + 1)^n/(1 + x*Ser(A)^n)^(n+1) ),#A)); polcoeff(Ser(A),n)}

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

%K sign

%O 0,4

%A _Paul D. Hanna_, Mar 12 2019