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%I #5 May 21 2023 00:44:19
%S 1,5,27,155,929,5730,36083,230935,1497739,9822060,65021849,433937545,
%T 2916359840,19720710150,134078691289,915994242780,6284957607075,
%U 43291450899490,299248617182754,2075172105905550,14432704539830007,100648564848019045,703624464015723819
%N Expansion of g.f. A(x) satisfying 5 = Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(2*n-1))^(n+1).
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
%F (1) 5 = Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(2*n-1))^(n+1).
%F (2) 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 + 5*A(x)*x^(2*n+1))^(n-1).
%F (3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(2*n-1))^n.
%F (4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (5*A(x) + x^(2*n-1))^(n-1).
%F (5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n^2) / (1 + 5*A(x)*x^(2*n+1))^n.
%F a(n) = Sum_{k=0..floor(n/2)} A359670(n-k,n-2*k) * 5^(n-2*k) for n >= 0.
%e G.f.: A(x) = 1 + 5*x + 27*x^2 + 155*x^3 + 929*x^4 + 5730*x^5 + 36083*x^6 + 230935*x^7 + 1497739*x^8 + 9822060*x^9 + 65021849*x^10 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
%o A[#A] = polcoeff(1 - sum(m=-#A, #A, (-1)^m * x^m * (5*Ser(A) + x^(2*m-1))^(m+1) ), #A-1)/5); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A363142, A363182, A363183, A363184.
%Y Cf. A359670.
%K nonn
%O 0,2
%A _Paul D. Hanna_, May 20 2023
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