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A379205 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 5)^(n+1).

%I A379205 #7 Dec 20 2024 23:39:04
%S A379205 1,7,74,998,15268,251427,4345869,77751128,1427455842,26740178711,
%T A379205 509068777424,9820550568868,191554931918517,3771529984556599,
%U A379205 74857068226445132,1496158969938529383,30086862802675119068,608303992207446069349,12358069554479794052292,252144178158939689795128
%N A379205 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 5)^(n+1).
%H A379205 Paul D. Hanna, <a href="/A379205/b379205.txt">Table of n, a(n) for n = 1..230</a>
%F A379205 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A379205 (1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 5)^(n+1).
%F A379205 (2) 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n - 5)^n.
%F A379205 (3) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + 5*A(x)^(n+1))^n.
%F A379205 (4) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 - 5*A(x)^(n+1))^(n+1).
%F A379205 (5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + 5)^(n+1) ).
%e A379205 G.f.: A(x) = x + 7*x^2 + 74*x^3 + 998*x^4 + 15268*x^5 + 251427*x^6 + 4345869*x^7 + 77751128*x^8 + 1427455842*x^9 + 26740178711*x^10 + ...
%o A379205 (PARI) {a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
%o A379205 V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + 5)^(m+1) ), #V-3); ); polcoef(A, n)}
%o A379205 for(n=1, 40, print1(a(n), ", "))
%Y A379205 Cf. A379200, A379199, A166952, A378264, A379202, A379203, A379204.
%K A379205 nonn
%O A379205 1,2
%A A379205 _Paul D. Hanna_, Dec 20 2024