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A357795 a(n) = coefficient of x^n in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^n * (1 - x^(n+2))^n * A(x)^(n+2).

%I A357795 #7 Dec 24 2022 11:16:46
%S A357795 1,4,26,300,4134,61696,969660,15837400,266125823,4571229248,
%T A357795 79904206064,1416736880104,25418030469904,460600399886240,
%U A357795 8417980252615072,154985730303047328,2871904782258356719,53519211809275995362,1002383232008661189884,18858606600633628740774
%N A357795 a(n) = coefficient of x^n in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^n * (1 - x^(n+2))^n * A(x)^(n+2).
%C A357795 Related identity: 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^(3*n) * (y - x^n)^(n-1), which holds formally for all y.
%H A357795 Paul D. Hanna, <a href="/A357795/b357795.txt">Table of n, a(n) for n = 0..200</a>
%F A357795 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
%F A357795 (1) 1 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^n * (1 - x^(n+2))^n * A(x)^(n+2).
%F A357795 (2) 1 = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)*(n-2)/3! * x^(n*(n-3)) / ((1 - x^(n-2))^n * A(x)^(n-2)).
%e A357795 G.f.: A(x) = 1 + 4*x + 26*x^2 + 300*x^3 + 4134*x^4 + 61696*x^5 + 969660*x^6 + 15837400*x^7 + 266125823*x^8 + 4571229248*x^9 + 79904206064*x^10 + ...
%o A357795 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A357795 A[#A] = polcoeff( sum(n=-#A-1, #A+1, n*(n+1)*(n+2)/3! * x^n * if(n==-2,0, (1 - x^(n+2) +x*O(x^#A) )^n) * Ser(A)^(n+2) ), #A-1) ); H=A; A[n+1]}
%o A357795 for(n=0, 30, print1(a(n), ", "))
%o A357795 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A357795 A[#A] = polcoeff( sum(n=-#A-1, #A+1, (-1)^(n-1) * n*(n-1)*(n-2)/3! * x^(n*(n-3)) * if(n==2,0, 1/(1 - x^(n-2) +x*O(x^#A) )^n) / Ser(A)^(n-2) ), #A-1) ); A[n+1]}
%o A357795 for(n=0, 30, print1(a(n), ", "))
%Y A357795 Cf. A357158, A357794, A357796.
%K A357795 nonn
%O A357795 0,2
%A A357795 _Paul D. Hanna_, Dec 22 2022