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A358965 a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(6*n+5))^(n-1).

%I A358965 #16 Dec 09 2022 14:34:47
%S A358965 1,2,7,30,143,729,3876,21321,120195,690816,4032807,23846485,142530516,
%T A358965 859719414,5226571568,31992109155,197002217763,1219554190530,
%U A358965 7585453430037,47380560231549,297081856642195,1869191995298989,11797744585161792,74678247991840230,473954364916279312
%N A358965 a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(6*n+5))^(n-1).
%C A358965 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(6*n+5))^n, which holds formally for all y.
%H A358965 Paul D. Hanna, <a href="/A358965/b358965.txt">Table of n, a(n) for n = 0..200</a>
%F A358965 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A358965 (1) 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(6*n+5))^(n-1).
%F A358965 (2) x^5 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(6*n^2) / (1 - x^(6*n-5)*A(x))^(n+1).
%F A358965 (3) A(x) = Sum_{n=-oo..+oo} x^(7*n+5)* (A(x) - x^(6*n+5))^(n-1).
%F A358965 (4) A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(6*n*(n-1)) / (1 - x^(6*n-5)*A(x))^(n+1).
%F A358965 (5) 0 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(6*n*(n-1)) / (1 - x^(6*n-5)*A(x))^n.
%e A358965 G.f.: A(x) = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 729*x^5 + 3876*x^6 + 21321*x^7 + 120195*x^8 + 690816*x^9 + 4032807*x^10 + ...
%e A358965 where A = A(x) satisfies the doubly infinite sum
%e A358965 1 = ... + x^(-2)*(A - x^(-7))^(-3) + x^(-1)*(A - x^(-1))^(-2) + (A - x^5)^(-1) + x*(A - x^11)^0 + x^2*(A - x^17) + x^3*(A - x^23)^2 + x^4*(A - x^29)^3 + ... + x^n * (A - x^(6*n+5))^(n-1) + ...
%e A358965 also,
%e A358965 A(x) = ... + x^72/(1 - x^(-23)*A)^(-2) - x^36/(1 - x^(-17)*A)^(-1) + x^12 - 1/(1 - x^(-5)*A) + 1/(1 - x*A)^2 - x^12/(1 - x^7*A)^3 + x^36/(1 - x^13*A)^4 - x^72/(1 - x^19*A)^5 + ... + (-1)^(n+1)*x^(6*n*(n-1))/(1 - x^(6*n-5)*A)^(n+1) + ...
%o A358965 (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A358965 A[#A] = polcoeff( sum(n=-#A,#A, x^n * (Ser(A) - x^(6*n+5))^(n-1) ), #A-1) );A[n+1]}
%o A358965 for(n=0,30,print1(a(n),", "))
%Y A358965 Cf. A358961, A358962, A358963, A358964.
%K A358965 nonn
%O A358965 0,2
%A A358965 _Paul D. Hanna_, Dec 07 2022