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A357797 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (2 + x^n)^n * A(x)^n.

%I A357797 #15 Mar 14 2023 18:37:40
%S A357797 1,1,5,18,85,374,1659,7774,36876,177494,867424,4285653,21373782,
%T A357797 107475746,544244911,2773091748,14207171278,73140904609,378184133959,
%U A357797 1963127909395,10226682384980,53446907352828,280150058149086,1472424136948438,7758105323877698,40970959715619200,216830651728330127
%N A357797 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (2 + x^n)^n * A(x)^n.
%C A357797 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.
%H A357797 Paul D. Hanna, <a href="/A357797/b357797.txt">Table of n, a(n) for n = 0..300</a>
%F A357797 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
%F A357797 (1) x = Sum_{n=-oo..+oo} (-1)^n * x^n * (2 + x^n)^n * A(x)^n.
%F A357797 (2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ((1 + 2*x^n)^n * A(x)^n).
%F A357797 (3) a(n) = Sum_{k=0..floor(2*n/3)} A359720(n,k)*2^k, for n >= 0.
%e A357797 G.f.: A(x) = 1 + x + 5*x^2 + 18*x^3 + 85*x^4 + 374*x^5 + 1659*x^6 + 7774*x^7 + 36876*x^8 + 177494*x^9 + 867424*x^10 + 4285653*x^11 + 21373782*x^12 + ...
%o A357797 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A357797 A[#A] = polcoeff(x - sum(n=-#A-1, #A+1, (-1)^n * x^n * (2 + x^n +x*O(x^#A) )^n * Ser(A)^n ), #A-1) ); A[n+1]}
%o A357797 for(n=0, 30, print1(a(n), ", "))
%o A357797 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A357797 A[#A] = polcoeff(x - sum(n=-#A, #A, (-1)^n * x^(n*(n-1)) / ((1 + 2*x^n +x*O(x^#A) )^n * Ser(A)^n) ), #A-1) ); A[n+1]}
%o A357797 for(n=0, 30, print1(a(n), ", "))
%Y A357797 Cf. A359720, A359721, A359723, A357798.
%K A357797 nonn
%O A357797 0,3
%A A357797 _Paul D. Hanna_, Dec 22 2022